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Author SHA1 Message Date
Connor Johnstone
e1e6f8b4bb Finished bs3 (at least for now) 2025-10-24 10:32:32 -04:00
Connor Johnstone
bd6f3b8ee4 Fixed things to use cubic interpolation and tests pass 2025-10-23 17:17:22 -04:00
Connor Johnstone
500bbfcf86 Initial attempt at bs3. Tests not passing yet 2025-10-23 16:56:48 -04:00
Connor Johnstone
e3788bf607 Added the roadmap 2025-10-23 16:47:48 -04:00
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4
Cargo.toml
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@@ -27,3 +27,7 @@ harness = false
[[bench]] [[bench]]
name = "orbit" name = "orbit"
harness = false harness = false
[[bench]]
name = "bs3_vs_dp5"
harness = false

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# BS3 vs DP5 Benchmark Results
Generated: 2025-10-23
## Summary
Comprehensive performance comparison between **BS3** (Bogacki-Shampine 3rd order) and **DP5** (Dormand-Prince 5th order) integrators across various test problems and tolerances.
## Key Findings
### Overall Performance Comparison
**DP5 is consistently faster than BS3 across all tested scenarios**, typically by a factor of **1.5x to 4.3x**.
This might seem counterintuitive since BS3 uses fewer stages (4 vs 7), but several factors explain DP5's superior performance:
1. **Higher Order = Larger Steps**: DP5's 5th order accuracy allows larger timesteps while maintaining the same error tolerance
2. **Optimized Implementation**: DP5 has been highly optimized in the existing codebase
3. **Smoother Problems**: The test problems are relatively smooth, favoring higher-order methods
### When to Use BS3
Despite being slower in these benchmarks, BS3 still has value:
- **Lower memory overhead**: Simpler dense output (4 values vs 5 for DP5)
- **Moderate accuracy needs**: For tolerances around 1e-3 to 1e-5 where speed difference is smaller
- **Educational/algorithmic diversity**: Different method characteristics
- **Specific problem types**: May perform better on less smooth or oscillatory problems
## Detailed Results
### 1. Exponential Decay (`y' = -0.5y`, tolerance 1e-5)
| Method | Time | Ratio |
|--------|------|-------|
| **BS3** | 3.28 µs | 1.92x slower |
| **DP5** | 1.70 µs | baseline |
Simple 1D problem with smooth exponential solution.
### 2. Harmonic Oscillator (`y'' + y = 0`, tolerance 1e-5)
| Method | Time | Ratio |
|--------|------|-------|
| **BS3** | 30.70 µs | 2.25x slower |
| **DP5** | 13.67 µs | baseline |
2D conservative system with periodic solution.
### 3. Nonlinear Pendulum (tolerance 1e-6)
| Method | Time | Ratio |
|--------|------|-------|
| **BS3** | 132.35 µs | 3.57x slower |
| **DP5** | 37.11 µs | baseline |
Nonlinear 2D system with trigonometric terms.
### 4. Orbital Mechanics (6D, tolerance 1e-6)
| Method | Time | Ratio |
|--------|------|-------|
| **BS3** | 124.72 µs | 1.45x slower |
| **DP5** | 86.10 µs | baseline |
Higher-dimensional problem with gravitational dynamics.
### 5. Interpolation Performance
| Method | Time (solve + 100 interpolations) | Ratio |
|--------|-----------------------------------|-------|
| **BS3** | 19.68 µs | 4.81x slower |
| **DP5** | 4.09 µs | baseline |
BS3 uses cubic Hermite interpolation, DP5 uses optimized 5th order interpolation.
### 6. Tolerance Scaling
Performance across different tolerance levels (`y' = -y` problem):
| Tolerance | BS3 Time | DP5 Time | Ratio (BS3/DP5) |
|-----------|----------|----------|-----------------|
| 1e-3 | 1.63 µs | 1.26 µs | 1.30x |
| 1e-4 | 2.61 µs | 1.54 µs | 1.70x |
| 1e-5 | 4.64 µs | 2.03 µs | 2.28x |
| 1e-6 | 8.76 µs | ~2.6 µs* | ~3.4x* |
| 1e-7 | -** | -** | - |
\* Estimated from trend (benchmark timed out)
\** Not completed
**Observation**: The performance gap widens as tolerance tightens, because DP5's higher order allows it to take larger steps while maintaining accuracy.
## Conclusions
### Performance Characteristics
1. **DP5 is the better default choice** for most problems requiring moderate to high accuracy
2. **Performance gap increases** with tighter tolerances (favoring DP5)
3. **Higher dimensions** slightly favor BS3 relative to DP5 (1.45x vs 3.57x slowdown)
4. **Interpolation** strongly favors DP5 (4.8x faster)
### Implementation Quality
Both integrators pass all accuracy and convergence tests:
- ✅ BS3: 3rd order convergence rate verified
- ✅ DP5: 5th order convergence rate verified (existing implementation)
- ✅ Both: FSAL property correctly implemented
- ✅ Both: Dense output accurate to specified order
### Future Optimizations
Potential improvements to BS3 performance:
1. **Specialized dense output**: Implement the optimized BS3 interpolation from the 1996 paper
2. **SIMD optimization**: Vectorize stage computations
3. **Memory layout**: Optimize cache usage for k-value storage
4. **Inline hints**: Add compiler hints for critical paths
Even with optimizations, DP5 will likely remain faster for these problem types due to its higher order.
## Recommendations
- **Use DP5**: For general-purpose ODE solving, especially for smooth problems
- **Use BS3**: When you specifically need:
- Lower memory usage
- A 3rd order reference implementation
- Comparison with other 3rd order methods
## Methodology
- **Tool**: Criterion.rs v0.7.0
- **Samples**: 100 per benchmark
- **Warmup**: 3 seconds per benchmark
- **Optimization**: Release mode with full optimizations
- **Platform**: Linux x86_64
- **Compiler**: rustc (specific version from build)
All benchmarks use `std::hint::black_box()` to prevent compiler optimizations from affecting timing.
## Reproducing Results
```bash
cargo bench --bench bs3_vs_dp5
```
Detailed plots and statistics are available in `target/criterion/`.

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# Benchmarks
This directory contains performance benchmarks for the ODE solver library.
## Running Benchmarks
To run all benchmarks:
```bash
cargo bench
```
To run a specific benchmark file:
```bash
cargo bench --bench bs3_vs_dp5
cargo bench --bench simple_1d
cargo bench --bench orbit
```
## Benchmark Suites
### `bs3_vs_dp5.rs` - BS3 vs DP5 Comparison
Comprehensive performance comparison between the Bogacki-Shampine 3(2) method (BS3) and Dormand-Prince 4(5) method (DP5).
**Test Problems:**
1. **Exponential Decay** - Simple 1D problem: `y' = -0.5*y`
2. **Harmonic Oscillator** - 2D conservative system: `y'' + y = 0`
3. **Nonlinear Pendulum** - Nonlinear 2D system with trigonometric terms
4. **Orbital Mechanics** - 6D system with gravitational dynamics
5. **Interpolation** - Performance of dense output interpolation
6. **Tolerance Scaling** - How methods perform across tolerance ranges (1e-3 to 1e-7)
**Expected Results:**
- **BS3** should be faster for moderate tolerances (1e-3 to 1e-6) on simple problems
- Lower overhead: 4 stages vs 7 stages for DP5
- FSAL property: effective cost ~3 function evaluations per step
- **DP5** should be faster for tight tolerances (< 1e-7)
- Higher order allows larger steps
- Better for problems requiring high accuracy
- **Interpolation**: DP5 has more sophisticated interpolation, may be faster/more accurate
### `simple_1d.rs` - Simple 1D Problem
Basic benchmark for a simple 1D exponential decay problem using DP5.
### `orbit.rs` - Orbital Mechanics
6D orbital mechanics problem using DP5.
## Benchmark Results Interpretation
Criterion outputs timing statistics for each benchmark:
- **Time**: Mean execution time with confidence interval
- **Outliers**: Number of measurements significantly different from the mean
- **Plots**: Stored in `target/criterion/` (if gnuplot is available)
### Performance Comparison
When comparing BS3 vs DP5:
1. **For moderate accuracy (tol ~ 1e-5)**:
- BS3 typically uses ~1.5-2x the time per problem
- But this can vary by problem characteristics
2. **For high accuracy (tol ~ 1e-7)**:
- DP5 becomes more competitive or faster
- Higher order allows fewer steps
3. **Memory usage**:
- BS3: Stores 4 values for dense output [y0, y1, f0, f1]
- DP5: Stores 5 values for dense output [rcont1..rcont5]
- Difference is minimal for most problems
## Notes
- Benchmarks use `std::hint::black_box()` to prevent compiler optimizations
- Each benchmark runs multiple iterations to get statistically significant results
- Results may vary based on:
- System load
- CPU frequency scaling
- Compiler optimizations
- Problem characteristics (stiffness, nonlinearity, dimension)
## Adding New Benchmarks
To add a new benchmark:
1. Create a new file in `benches/` (e.g., `my_benchmark.rs`)
2. Add benchmark configuration to `Cargo.toml`:
```toml
[[bench]]
name = "my_benchmark"
harness = false
```
3. Use the Criterion framework:
```rust
use criterion::{criterion_group, criterion_main, Criterion};
use std::hint::black_box;
fn my_bench(c: &mut Criterion) {
c.bench_function("my_test", |b| {
b.iter(|| {
black_box({
// Code to benchmark
});
});
});
}
criterion_group!(benches, my_bench);
criterion_main!(benches);
```

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use criterion::{criterion_group, criterion_main, BenchmarkId, Criterion};
use nalgebra::{Vector1, Vector2, Vector6};
use ordinary_diffeq::prelude::*;
use std::f64::consts::PI;
use std::hint::black_box;
// Simple 1D exponential decay problem
// y' = -k*y, y(0) = 1
fn bench_exponential_decay(c: &mut Criterion) {
type Params = (f64,);
let params = (0.5,);
fn derivative(_t: f64, y: Vector1<f64>, p: &Params) -> Vector1<f64> {
Vector1::new(-p.0 * y[0])
}
let y0 = Vector1::new(1.0);
let controller = PIController::default();
let mut group = c.benchmark_group("exponential_decay");
// Moderate tolerance - where BS3 should excel
let tol = 1e-5;
group.bench_function("bs3_tol_1e-5", |b| {
let ode = ODE::new(&derivative, 0.0, 10.0, y0, params);
let bs3 = BS3::new().a_tol(tol).r_tol(tol);
b.iter(|| {
black_box({
Problem::new(ode, bs3, controller).solve();
});
});
});
group.bench_function("dp5_tol_1e-5", |b| {
let ode = ODE::new(&derivative, 0.0, 10.0, y0, params);
let dp45 = DormandPrince45::new().a_tol(tol).r_tol(tol);
b.iter(|| {
black_box({
Problem::new(ode, dp45, controller).solve();
});
});
});
group.finish();
}
// 2D harmonic oscillator
// y'' + y = 0, or as system: y1' = y2, y2' = -y1
fn bench_harmonic_oscillator(c: &mut Criterion) {
type Params = ();
fn derivative(_t: f64, y: Vector2<f64>, _p: &Params) -> Vector2<f64> {
Vector2::new(y[1], -y[0])
}
let y0 = Vector2::new(1.0, 0.0);
let controller = PIController::default();
let mut group = c.benchmark_group("harmonic_oscillator");
let tol = 1e-5;
group.bench_function("bs3_tol_1e-5", |b| {
let ode = ODE::new(&derivative, 0.0, 20.0, y0, ());
let bs3 = BS3::new().a_tol(tol).r_tol(tol);
b.iter(|| {
black_box({
Problem::new(ode, bs3, controller).solve();
});
});
});
group.bench_function("dp5_tol_1e-5", |b| {
let ode = ODE::new(&derivative, 0.0, 20.0, y0, ());
let dp45 = DormandPrince45::new().a_tol(tol).r_tol(tol);
b.iter(|| {
black_box({
Problem::new(ode, dp45, controller).solve();
});
});
});
group.finish();
}
// Nonlinear pendulum
// theta'' + (g/L)*sin(theta) = 0
fn bench_pendulum(c: &mut Criterion) {
type Params = (f64, f64); // (g, L)
let params = (9.81, 1.0);
fn derivative(_t: f64, y: Vector2<f64>, p: &Params) -> Vector2<f64> {
let &(g, l) = p;
let theta = y[0];
let d_theta = y[1];
Vector2::new(d_theta, -(g / l) * theta.sin())
}
let y0 = Vector2::new(0.0, PI / 2.0); // Start from rest at angle 0, velocity PI/2
let controller = PIController::default();
let mut group = c.benchmark_group("pendulum");
let tol = 1e-6;
group.bench_function("bs3_tol_1e-6", |b| {
let ode = ODE::new(&derivative, 0.0, 10.0, y0, params);
let bs3 = BS3::new().a_tol(tol).r_tol(tol);
b.iter(|| {
black_box({
Problem::new(ode, bs3, controller).solve();
});
});
});
group.bench_function("dp5_tol_1e-6", |b| {
let ode = ODE::new(&derivative, 0.0, 10.0, y0, params);
let dp45 = DormandPrince45::new().a_tol(tol).r_tol(tol);
b.iter(|| {
black_box({
Problem::new(ode, dp45, controller).solve();
});
});
});
group.finish();
}
// 6D orbital mechanics - higher dimensional problem
fn bench_orbit_6d(c: &mut Criterion) {
let mu = 3.98600441500000e14;
type Params = (f64,);
let params = (mu,);
fn derivative(_t: f64, state: Vector6<f64>, p: &Params) -> Vector6<f64> {
let acc = -(p.0 * state.fixed_rows::<3>(0)) / (state.fixed_rows::<3>(0).norm().powi(3));
Vector6::new(state[3], state[4], state[5], acc[0], acc[1], acc[2])
}
let y0 = Vector6::new(
4.263868426884883e6,
5.146189057155391e6,
1.1310208421331816e6,
-5923.454461876975,
4496.802639690076,
1870.3893008991558,
);
let controller = PIController::new(0.37, 0.04, 10.0, 0.2, 1000.0, 0.9, 0.01);
let mut group = c.benchmark_group("orbit_6d");
// Test at moderate tolerance
let tol = 1e-6;
group.bench_function("bs3_tol_1e-6", |b| {
let ode = ODE::new(&derivative, 0.0, 10000.0, y0, params);
let bs3 = BS3::new().a_tol(tol).r_tol(tol);
b.iter(|| {
black_box({
Problem::new(ode, bs3, controller).solve();
});
});
});
group.bench_function("dp5_tol_1e-6", |b| {
let ode = ODE::new(&derivative, 0.0, 10000.0, y0, params);
let dp45 = DormandPrince45::new().a_tol(tol).r_tol(tol);
b.iter(|| {
black_box({
Problem::new(ode, dp45, controller).solve();
});
});
});
group.finish();
}
// Benchmark interpolation performance
fn bench_interpolation(c: &mut Criterion) {
type Params = ();
fn derivative(_t: f64, y: Vector1<f64>, _p: &Params) -> Vector1<f64> {
Vector1::new(y[0])
}
let y0 = Vector1::new(1.0);
let controller = PIController::default();
let mut group = c.benchmark_group("interpolation");
let tol = 1e-6;
// BS3 with interpolation
group.bench_function("bs3_with_interpolation", |b| {
let ode = ODE::new(&derivative, 0.0, 5.0, y0, ());
let bs3 = BS3::new().a_tol(tol).r_tol(tol);
b.iter(|| {
black_box({
let solution = Problem::new(ode, bs3, controller).solve();
// Interpolate at 100 points
let _: Vec<_> = (0..100).map(|i| solution.interpolate(i as f64 * 0.05)).collect();
});
});
});
// DP5 with interpolation
group.bench_function("dp5_with_interpolation", |b| {
let ode = ODE::new(&derivative, 0.0, 5.0, y0, ());
let dp45 = DormandPrince45::new().a_tol(tol).r_tol(tol);
b.iter(|| {
black_box({
let solution = Problem::new(ode, dp45, controller).solve();
// Interpolate at 100 points
let _: Vec<_> = (0..100).map(|i| solution.interpolate(i as f64 * 0.05)).collect();
});
});
});
group.finish();
}
// Tolerance scaling benchmark - how do methods perform at different tolerances?
fn bench_tolerance_scaling(c: &mut Criterion) {
type Params = ();
fn derivative(_t: f64, y: Vector1<f64>, _p: &Params) -> Vector1<f64> {
Vector1::new(-y[0])
}
let y0 = Vector1::new(1.0);
let controller = PIController::default();
let mut group = c.benchmark_group("tolerance_scaling");
let tolerances = [1e-3, 1e-4, 1e-5, 1e-6, 1e-7];
for &tol in &tolerances {
group.bench_with_input(BenchmarkId::new("bs3", tol), &tol, |b, &tol| {
let ode = ODE::new(&derivative, 0.0, 10.0, y0, ());
let bs3 = BS3::new().a_tol(tol).r_tol(tol);
b.iter(|| {
black_box({
Problem::new(ode, bs3, controller).solve();
});
});
});
group.bench_with_input(BenchmarkId::new("dp5", tol), &tol, |b, &tol| {
let ode = ODE::new(&derivative, 0.0, 10.0, y0, ());
let dp45 = DormandPrince45::new().a_tol(tol).r_tol(tol);
b.iter(|| {
black_box({
Problem::new(ode, dp45, controller).solve();
});
});
});
}
group.finish();
}
criterion_group!(
benches,
bench_exponential_decay,
bench_harmonic_oscillator,
bench_pendulum,
bench_orbit_6d,
bench_interpolation,
bench_tolerance_scaling,
);
criterion_main!(benches);

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# Feature File Templates
This document contains brief summaries for features 6-38. Detailed feature files should be created when you're ready to implement each one, using the detailed examples in features 01-05 and 12 as templates.
## How to Use This Document
When ready to implement a feature:
1. Copy the template structure from features/01-bs3-method.md or similar
2. Fill in the details from the summary below
3. Add implementation-specific details
4. Create comprehensive testing requirements
---
## Feature 06: CallbackSet
**Description**: Compose multiple callbacks with ordering
**Dependencies**: Discrete callbacks
**Effort**: Small
**Key Points**: Builder pattern, execution priority, enable/disable
## Feature 07: Saveat Functionality
**Description**: Save solution at specific timepoints
**Dependencies**: None
**Effort**: Medium
**Key Points**: Interpolation to exact times, dense vs sparse saving, memory efficiency
## Feature 08: Solution Derivatives
**Description**: Access derivatives at any time via interpolation
**Dependencies**: None
**Effort**: Small
**Key Points**: `solution.derivative(t)` interface, use dense output or finite differences
## Feature 09: DP8 Method
**Description**: Dormand-Prince 8th order method
**Dependencies**: None
**Effort**: Medium
**Key Points**: 13 stages, very high accuracy, tableau from literature
## Feature 10: FBDF Method
**Description**: Fixed-leading-coefficient BDF multistep method
**Dependencies**: Linear solver, Nordsieck representation
**Effort**: Large
**Key Points**: Variable order (1-5), excellent for very stiff problems, complex state management
## Feature 11: Rodas4/Rodas5P
**Description**: Higher-order Rosenbrock methods
**Dependencies**: Rosenbrock23
**Effort**: Medium
**Key Points**: 4th/5th order accuracy, more stages, better for higher accuracy stiff problems
## Feature 13: Default Algorithm Selection
**Description**: Smart defaults based on problem characteristics
**Dependencies**: Auto-switching, multiple algorithms
**Effort**: Medium
**Key Points**: Analyze tolerance, problem size, choose algorithms automatically
## Feature 14: Automatic Initial Stepsize
**Description**: Algorithm to compute good initial dt
**Dependencies**: None
**Effort**: Medium
**Key Points**: Based on Hairer & Wanner algorithm, uses local Lipschitz estimate
## Feature 15: PresetTimeCallback
**Description**: Callbacks at predetermined times
**Dependencies**: Discrete callbacks
**Effort**: Small
**Key Points**: Efficient time-based events, integration with tstops
## Feature 16: TerminateSteadyState
**Description**: Auto-detect when solution reaches steady state
**Dependencies**: Discrete callbacks
**Effort**: Small
**Key Points**: Monitor du/dt, terminate when small enough
## Feature 17: SavingCallback
**Description**: Custom saving logic beyond default
**Dependencies**: CallbackSet
**Effort**: Small
**Key Points**: User-defined save conditions, memory-efficient for large problems
## Feature 18: Linear Solver Infrastructure
**Description**: Generic linear solver interface and dense LU
**Dependencies**: None
**Effort**: Large
**Key Points**:
- Trait-based design for flexibility
- Dense LU factorization with partial pivoting
- Solve Ax = b efficiently
- Foundation for all implicit methods
- Consider using nalgebra's built-in LU or implement custom
## Feature 19: Jacobian Computation
**Description**: Finite difference and auto-diff Jacobians
**Dependencies**: None
**Effort**: Large
**Key Points**:
- Forward finite differences: (f(y+εe_j) - f(y))/ε
- Epsilon selection: √eps * max(|y_j|, 1)
- Sparse Jacobian support (future)
- Integration with AD crates (future)
## Feature 20: Low-Storage Runge-Kutta
**Description**: 2N/3N/4N storage variants for large systems
**Dependencies**: None
**Effort**: Medium
**Key Points**: Specialized RK methods that reuse storage, critical for PDEs via method-of-lines
## Feature 21: SSP Methods
**Description**: Strong Stability Preserving RK methods
**Dependencies**: None
**Effort**: Medium
**Key Points**: SSPRK22, SSPRK33, SSPRK43, SSPRK53, preserve TVD/monotonicity, for hyperbolic PDEs
## Feature 22: Symplectic Integrators
**Description**: Verlet, Leapfrog, KahanLi for Hamiltonian systems
**Dependencies**: None (second-order ODE support already exists)
**Effort**: Medium
**Key Points**: Preserve energy/symplectic structure, special for p,q formulation
## Feature 23: Verner Methods Suite
**Description**: Complete Verner family (Vern6, Vern8, Vern9)
**Dependencies**: Vern7
**Effort**: Medium
**Key Points**: Different orders for different accuracy needs, all highly efficient
## Feature 24: SDIRK Methods
**Description**: Singly Diagonally Implicit RK (KenCarp3/4/5)
**Dependencies**: Linear solver, nonlinear solver
**Effort**: Large
**Key Points**: IMEX methods, good for semi-stiff problems, L-stable
## Feature 25: Exponential Integrators
**Description**: Exp4, EPIRK4, EXPRB53 for semi-linear stiff
**Dependencies**: Matrix exponential computation
**Effort**: Large
**Key Points**: For du/dt = Lu + N(u), where L is linear stiff part
## Feature 26: Extrapolation Methods
**Description**: Richardson extrapolation with adaptive order
**Dependencies**: Linear solver
**Effort**: Large
**Key Points**: High accuracy from low-order methods, variable order selection
## Feature 27: Stabilized Methods
**Description**: ROCK2, ROCK4, RKC for mildly stiff
**Dependencies**: None
**Effort**: Medium
**Key Points**: Extended stability regions, good for PDEs with explicit time-stepping
## Feature 28: I Controller
**Description**: Basic integral controller
**Dependencies**: None
**Effort**: Small
**Key Points**: Simplest adaptive controller, mainly for comparison/testing
## Feature 29: Predictive Controller
**Description**: Advanced predictive step size control
**Dependencies**: None
**Effort**: Medium
**Key Points**: Predicts future error, more sophisticated than PID
## Feature 30: VectorContinuousCallback
**Description**: Multiple simultaneous event detection
**Dependencies**: CallbackSet
**Effort**: Medium
**Key Points**: More efficient than separate callbacks, shared root-finding
## Feature 31: PositiveDomain
**Description**: Enforce positivity constraints
**Dependencies**: CallbackSet
**Effort**: Small
**Key Points**: Ensures solution stays positive, important for physical systems
## Feature 32: ManifoldProjection
**Description**: Project solution onto constraint manifolds
**Dependencies**: CallbackSet
**Effort**: Medium
**Key Points**: For constrained mechanical systems, projection step after integration
## Feature 33: Nonlinear Solver Infrastructure
**Description**: Newton and quasi-Newton methods
**Dependencies**: Linear solver
**Effort**: Large
**Key Points**:
- Newton's method for implicit stages
- Line search
- Convergence criteria
- Foundation for SDIRK, FIRK methods
## Feature 34: Krylov Linear Solvers
**Description**: GMRES, BiCGStab for large sparse systems
**Dependencies**: Linear solver infrastructure
**Effort**: Large
**Key Points**: Iterative solvers for when LU factorization too expensive
## Feature 35: Preconditioners
**Description**: ILU, Jacobi, custom preconditioners
**Dependencies**: Krylov solvers
**Effort**: Large
**Key Points**: Accelerate Krylov methods, essential for large sparse systems
## Feature 36: FSAL Optimization
**Description**: First-Same-As-Last function evaluation reuse
**Dependencies**: None
**Effort**: Small
**Key Points**: Reduce function evaluations by ~14% for FSAL methods (DP5, Tsit5, etc.)
## Feature 37: Custom Norms
**Description**: User-definable error norms
**Dependencies**: None
**Effort**: Small
**Key Points**: L2, Linf, weighted norms, custom user functions
## Feature 38: Step/Stage Limiting
**Description**: Limit state values during integration
**Dependencies**: None
**Effort**: Small
**Key Points**: Enforce bounds on solution, prevent non-physical values
---
## Creating Detailed Feature Files
When you're ready to work on a feature, create a detailed file following this structure:
1. **Overview**: What is it, key characteristics
2. **Why This Feature Matters**: Motivation, use cases
3. **Dependencies**: What must be built first
4. **Implementation Approach**: Algorithm details, design decisions
5. **Implementation Tasks**: Detailed checklist with subtasks
6. **Testing Requirements**: Specific tests with expected results
7. **References**: Papers, Julia code, textbooks
8. **Complexity Estimate**: Effort and risk assessment
9. **Success Criteria**: How to know it's done right
10. **Future Enhancements**: What could be added later
See `features/01-bs3-method.md`, `features/02-vern7-method.md`, `features/03-rosenbrock23.md`, etc. for complete examples.

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# Getting Started with the Roadmap
This guide helps you navigate the development roadmap for the Rust ODE library.
## Roadmap Structure
```
roadmap/
├── README.md # Master overview with all features
├── GETTING_STARTED.md # This file
├── FEATURE_TEMPLATES.md # Brief summaries of features 6-38
└── features/
├── 01-bs3-method.md # Detailed implementation plan (example)
├── 02-vern7-method.md # Detailed implementation plan
├── 03-rosenbrock23.md # Detailed implementation plan
├── 04-pid-controller.md # Detailed implementation plan
├── 05-discrete-callbacks.md # Detailed implementation plan
├── 06-callback-set.md # Brief outline
└── 12-auto-switching.md # Detailed implementation plan
└── ... (create detailed files as needed)
```
## How to Use This Roadmap
### 1. Review the Master Plan
Start with `README.md` to see:
- All 38 planned features organized by tier
- Dependencies between features
- Current completion status
- Overall progress tracking
### 2. Choose Your Next Feature
**Recommended Order for Beginners:**
1. Start with Tier 1 features (essential)
2. Follow dependency chains
3. Mix difficulty levels (alternate hard and easy)
**Suggested First 5 Features:**
1. **BS3 Method** (feature #1) - Easy, builds confidence
2. **PID Controller** (feature #4) - Easy, immediate value
3. **Discrete Callbacks** (feature #5) - Easy, useful capability
4. **Vern7** (feature #2) - Medium, important algorithm
5. **Linear Solver Infrastructure** (feature #18) - Hard but foundational
### 3. Read the Detailed Feature File
Each detailed feature file contains:
- **Overview**: Quick introduction
- **Why It Matters**: Motivation
- **Dependencies**: What you need first
- **Implementation Approach**: Algorithm details
- **Implementation Tasks**: Detailed checklist
- **Testing Requirements**: How to verify it works
- **References**: Where to learn more
- **Complexity Estimate**: Time and difficulty
- **Success Criteria**: Definition of done
### 4. Implement the Feature
Follow the detailed task checklist:
- [ ] Read references and understand algorithm
- [ ] Implement core algorithm
- [ ] Write tests
- [ ] Document
- [ ] Benchmark
- [ ] Check off tasks as you complete them
### 5. Update the Roadmap
When you complete a feature:
1. Check the box in `README.md`
2. Update completion statistics
3. Note any lessons learned or deviations from plan
## Current State (Baseline)
Your library already has:
- ✅ Dormand-Prince 4(5) with dense output
- ✅ Tsit5 with dense output
- ✅ PI Controller
- ✅ Continuous callbacks with zero-crossing detection
- ✅ Solution interpolation interface
- ✅ Generic over compile-time array dimensions
- ✅ Support for second-order ODE problems
This is a solid foundation! The roadmap builds on this.
## Recommended Development Path
### Phase 1: Core Algorithm Diversity (Tier 1)
*Goal: Give users algorithm choices*
1. BS3 - Easy, quick win
2. Vern7 - High accuracy option
3. Build linear solver infrastructure
4. Rosenbrock23 - First stiff solver
5. PID Controller - Better adaptive stepping
6. Discrete Callbacks - More event types
**Milestone**: Can handle both non-stiff and stiff problems efficiently.
### Phase 2: Robustness & Automation (Tier 2)
*Goal: Make the library production-ready*
7. Auto-switching/stiffness detection
8. Automatic initial step size
9. More Rosenbrock methods (Rodas4)
10. BDF method
11. CallbackSet and advanced callbacks
12. Saveat functionality
**Milestone**: Library can solve most problems automatically with minimal user input.
### Phase 3: Specialization & Performance (Tier 3)
*Goal: Optimize for specific problem classes*
13. Low-storage RK for large systems
14. Symplectic integrators for Hamiltonian systems
15. SSP methods for hyperbolic PDEs
16. Verner suite completion
17. Advanced linear/nonlinear solvers
18. Performance optimizations (FSAL, custom norms)
**Milestone**: Best-in-class performance for specialized problem types.
## Development Tips
### Testing Strategy
Every feature should have:
1. **Convergence test**: Verify order of accuracy
2. **Correctness test**: Compare to known solutions
3. **Edge case tests**: Boundary conditions, error handling
4. **Benchmark**: Performance measurement
### Reference Material
When implementing a feature:
1. Read the Julia implementation for guidance
2. Check original papers for algorithm details
3. Verify tableau/coefficients from authoritative sources
4. Test against reference solutions from DiffEqDevDocs
### Common Pitfalls
- **Don't skip testing**: Numerical bugs are subtle
- **Verify tableau coefficients**: Transcription errors are common
- **Check interpolation**: Easy to get wrong
- **Test stiff problems**: If implementing stiff solvers
- **Benchmark early**: Performance problems easier to fix early
## Getting Help
### Resources
1. **Julia's OrdinaryDiffEq.jl**: Reference implementation
- Location: `/tmp/diffeq_copy/OrdinaryDiffEq.jl/`
- Well-tested, can compare behavior
2. **Hairer & Wanner textbooks**:
- "Solving ODEs I: Nonstiff Problems"
- "Solving ODEs II: Stiff and DAE Problems"
3. **DiffEqDevDocs**: Developer documentation
- https://docs.sciml.ai/DiffEqDevDocs/stable/
4. **Test problems**: Standard ODE test suite
- Van der Pol, Robertson, Pleiades, etc.
- Reference solutions available
### Creating New Detailed Feature Files
When ready to work on a feature that only has a brief summary:
1. Copy structure from `features/01-bs3-method.md`
2. Fill in details from `FEATURE_TEMPLATES.md`
3. Add algorithm-specific information
4. Create comprehensive task checklist
5. Define specific test requirements
6. Estimate complexity honestly
## Tracking Progress
### In README.md
Update the checkboxes as features are completed:
- [ ] Incomplete
- [x] Complete
Update completion statistics at bottom:
```
## Progress Tracking
Total Features: 38
- Tier 1: 8 features (3/8 complete) # Update these
- Tier 2: 12 features (0/12 complete)
- Tier 3: 18 features (0/18 complete)
```
### Optional: Keep a CHANGELOG.md
Document major milestones:
```markdown
# Changelog
## 2025-01-XX
- Completed BS3 method
- Completed PID controller
- Started Vern7 implementation
## 2025-01-YY
- Completed Vern7
- Linear solver infrastructure in progress
```
## Questions to Ask Before Starting
Before implementing a feature:
1. **Do I understand the algorithm?**
- Read the papers
- Understand the math
- Know the use cases
2. **Are dependencies satisfied?**
- Check the dependency list
- Make sure infrastructure exists
3. **Do I have test cases ready?**
- Know how to verify correctness
- Have reference solutions
4. **What's the success criteria?**
- Clear definition of "done"
- Performance targets
## Next Steps
1. Read `README.md` to see the full roadmap
2. Pick a feature to start with (suggest: BS3 or PID Controller)
3. Read its detailed feature file
4. Implement following the task checklist
5. Test thoroughly
6. Update the roadmap
7. Move to next feature!
Good luck! You're building something great. 🚀

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# Ordinary Differential Equations Library - Development Roadmap
This roadmap outlines the planned features for developing a comprehensive Rust-based ODE solver library, inspired by Julia's OrdinaryDiffEq.jl but adapted for Rust's strengths and idioms.
## Current Foundation
The library currently has:
- ✅ Dormand-Prince 4(5) adaptive explicit RK method with dense output
- ✅ Tsit5 explicit RK method with dense output
- ✅ PI step size controller
- ✅ Basic continuous callbacks with zero-crossing detection
- ✅ Solution interpolation interface
- ✅ Generic over array dimensions at compile-time
- ✅ Support for ordinary and second-order ODE problems
## Roadmap Organization
Features are organized into three tiers based on priority and dependencies:
- **Tier 1**: Essential features for general-purpose ODE solving
- **Tier 2**: Important features for robustness and broader applicability
- **Tier 3**: Advanced/specialized features for specific problem classes
Each feature below links to a detailed implementation plan in the `features/` directory.
---
## Tier 1: Essential Features
### Algorithms
- [x] **[BS3 (Bogacki-Shampine 3/2)](features/01-bs3-method.md)** ✅ COMPLETED
- 3rd order explicit RK method with 2nd order error estimate
- Good for moderate accuracy, lower cost than DP5
- **Dependencies**: None
- **Effort**: Small
- [ ] **[Vern7 (Verner 7th order)](features/02-vern7-method.md)**
- 7th order explicit RK method for high-accuracy non-stiff problems
- Efficient for tight tolerances
- **Dependencies**: None
- **Effort**: Medium
- [ ] **[Rosenbrock23](features/03-rosenbrock23.md)**
- L-stable 2nd/3rd order Rosenbrock-W method
- First working stiff solver
- **Dependencies**: Linear solver infrastructure, Jacobian computation
- **Effort**: Large
### Controllers
- [ ] **[PID Controller](features/04-pid-controller.md)**
- Proportional-Integral-Derivative step size controller
- Better stability than PI controller for difficult problems
- **Dependencies**: None
- **Effort**: Small
### Callbacks
- [ ] **[Discrete Callbacks](features/05-discrete-callbacks.md)**
- Event detection based on conditions (not zero-crossings)
- Useful for time-based events, iteration counts, etc.
- **Dependencies**: None
- **Effort**: Small
- [ ] **[CallbackSet](features/06-callback-set.md)**
- Compose multiple callbacks with ordering
- Essential for complex simulations
- **Dependencies**: Discrete callbacks
- **Effort**: Small
### Solution Interface
- [ ] **[Saveat Functionality](features/07-saveat.md)**
- Save solution at specific timepoints
- Dense vs sparse saving strategies
- **Dependencies**: None
- **Effort**: Medium
- [ ] **[Solution Derivatives](features/08-solution-derivatives.md)**
- Access derivatives at any time point via interpolation
- `solution.derivative(t)` interface
- **Dependencies**: None
- **Effort**: Small
---
## Tier 2: Important for Robustness
### Algorithms
- [ ] **[DP8 (Dormand-Prince 8th order)](features/09-dp8-method.md)**
- 8th order explicit RK for very high accuracy
- Complements Vern7 for algorithm selection
- **Dependencies**: None
- **Effort**: Medium
- [ ] **[FBDF (Fixed-leading-coefficient BDF)](features/10-fbdf-method.md)**
- Multistep method for very stiff problems
- More robust than Rosenbrock for some problem classes
- **Dependencies**: Linear solver infrastructure, Nordsieck representation
- **Effort**: Large
- [ ] **[Rodas4/Rodas5P](features/11-rodas-methods.md)**
- Higher-order Rosenbrock methods (4th/5th order)
- Better accuracy for stiff problems
- **Dependencies**: Rosenbrock23
- **Effort**: Medium
### Algorithm Selection
- [ ] **[Auto-Switching / Stiffness Detection](features/12-auto-switching.md)**
- Automatic detection of stiffness
- Switch between non-stiff and stiff solvers
- Composite algorithm infrastructure
- **Dependencies**: At least one stiff solver (Rosenbrock23 or FBDF)
- **Effort**: Large
- [ ] **[Default Algorithm Selection](features/13-default-algorithm.md)**
- Smart defaults based on problem characteristics
- `solve(problem)` without specifying algorithm
- **Dependencies**: Auto-switching, multiple algorithms
- **Effort**: Medium
### Initialization
- [ ] **[Automatic Initial Step Size](features/14-initial-stepsize.md)**
- Algorithm to determine good initial dt
- Based on local Lipschitz estimate
- **Dependencies**: None
- **Effort**: Medium
### Callbacks
- [ ] **[PresetTimeCallback](features/15-preset-time-callback.md)**
- Trigger callbacks at specific predetermined times
- Important for time-varying forcing functions
- **Dependencies**: Discrete callbacks
- **Effort**: Small
- [ ] **[TerminateSteadyState](features/16-terminate-steady-state.md)**
- Auto-detect when solution reaches steady state
- Stop integration early
- **Dependencies**: Discrete callbacks
- **Effort**: Small
- [ ] **[SavingCallback](features/17-saving-callback.md)**
- Custom saving logic beyond default
- For memory-efficient large-scale simulations
- **Dependencies**: CallbackSet
- **Effort**: Small
### Infrastructure
- [ ] **[Linear Solver Infrastructure](features/18-linear-solver-infrastructure.md)**
- Generic linear solver interface
- Dense LU factorization
- Foundation for implicit methods
- **Dependencies**: None
- **Effort**: Large
- [ ] **[Jacobian Computation](features/19-jacobian-computation.md)**
- Finite difference Jacobians
- Forward-mode automatic differentiation
- Sparse Jacobian support
- **Dependencies**: None
- **Effort**: Large
---
## Tier 3: Advanced & Specialized
### Algorithms
- [ ] **[Low-Storage Runge-Kutta](features/20-low-storage-rk.md)**
- 2N, 3N, 4N storage variants
- Critical for large-scale problems
- **Dependencies**: None
- **Effort**: Medium
- [ ] **[SSP (Strong Stability Preserving) Methods](features/21-ssp-methods.md)**
- SSPRK22, SSPRK33, SSPRK43, SSPRK53, etc.
- For hyperbolic PDEs and method-of-lines
- **Dependencies**: None
- **Effort**: Medium
- [ ] **[Symplectic Integrators](features/22-symplectic-integrators.md)**
- Velocity Verlet, Leapfrog, KahanLi6, KahanLi8
- For Hamiltonian systems, preserves energy
- **Dependencies**: Second-order ODE infrastructure (already exists)
- **Effort**: Medium
- [ ] **[Verner Methods Suite](features/23-verner-methods.md)**
- Vern6, Vern8, Vern9
- Complete Verner family for different accuracy needs
- **Dependencies**: Vern7
- **Effort**: Medium
- [ ] **[SDIRK Methods](features/24-sdirk-methods.md)**
- KenCarp3, KenCarp4, KenCarp5
- Singly Diagonally Implicit RK for stiff problems
- **Dependencies**: Linear solver infrastructure, nonlinear solver
- **Effort**: Large
- [ ] **[Exponential Integrators](features/25-exponential-integrators.md)**
- Exp4, EPIRK4, EXPRB53
- For semi-linear stiff problems
- **Dependencies**: Matrix exponential computation
- **Effort**: Large
- [ ] **[Extrapolation Methods](features/26-extrapolation-methods.md)**
- Implicit Euler/Midpoint with Richardson extrapolation
- Adaptive order selection
- **Dependencies**: Linear solver infrastructure
- **Effort**: Large
- [ ] **[Stabilized Methods](features/27-stabilized-methods.md)**
- ROCK2, ROCK4, RKC
- For mildly stiff problems with large systems
- **Dependencies**: None
- **Effort**: Medium
### Controllers
- [ ] **[I Controller](features/28-i-controller.md)**
- Basic integral controller
- For completeness and testing
- **Dependencies**: None
- **Effort**: Small
- [ ] **[Predictive Controller](features/29-predictive-controller.md)**
- Advanced controller with prediction
- For challenging adaptive stepping scenarios
- **Dependencies**: None
- **Effort**: Medium
### Advanced Callbacks
- [ ] **[VectorContinuousCallback](features/30-vector-continuous-callback.md)**
- Multiple simultaneous event detection
- More efficient than multiple callbacks
- **Dependencies**: CallbackSet
- **Effort**: Medium
- [ ] **[PositiveDomain](features/31-positive-domain.md)**
- Enforce positivity constraints
- Important for physical systems
- **Dependencies**: CallbackSet
- **Effort**: Small
- [ ] **[ManifoldProjection](features/32-manifold-projection.md)**
- Project solution onto constraint manifolds
- For constrained mechanical systems
- **Dependencies**: CallbackSet
- **Effort**: Medium
### Infrastructure
- [ ] **[Nonlinear Solver Infrastructure](features/33-nonlinear-solver.md)**
- Newton's method
- Quasi-Newton methods
- Generic nonlinear solver interface
- **Dependencies**: Linear solver infrastructure
- **Effort**: Large
- [ ] **[Krylov Linear Solvers](features/34-krylov-solvers.md)**
- GMRES, BiCGStab
- For large sparse systems
- **Dependencies**: Linear solver infrastructure
- **Effort**: Large
- [ ] **[Preconditioners](features/35-preconditioners.md)**
- ILU, Jacobi, custom preconditioners
- Accelerate Krylov methods
- **Dependencies**: Krylov solvers
- **Effort**: Large
### Performance & Optimization
- [ ] **[FSAL Optimization](features/36-fsal-optimization.md)**
- First-Same-As-Last reuse
- Reduce function evaluations
- **Dependencies**: None
- **Effort**: Small
- [ ] **[Custom Norms](features/37-custom-norms.md)**
- User-definable error norms
- L2, Linf, weighted norms
- **Dependencies**: None
- **Effort**: Small
- [ ] **[Step/Stage Limiting](features/38-step-stage-limiting.md)**
- Limit state values during integration
- For bounded problems
- **Dependencies**: None
- **Effort**: Small
---
## Implementation Notes
### General Principles
1. **Rust-first design**: Leverage Rust's type system, zero-cost abstractions, and safety guarantees
2. **Compile-time optimization**: Use const generics for array sizes where beneficial
3. **Trait-based abstraction**: Generic over array types, number types, and algorithm components
4. **Comprehensive testing**: Each feature needs convergence tests and comparison to known solutions
5. **Benchmarking**: Track performance as features are added
### Testing Strategy
Each algorithm implementation should include:
- **Convergence tests**: Verify order of accuracy
- **Correctness tests**: Compare to analytical solutions
- **Stiffness tests**: For stiff solvers, test on Van der Pol, Robertson, etc.
- **Callback tests**: Verify event detection accuracy
- **Regression tests**: Prevent performance degradation
### Documentation Requirements
- Public API documentation
- Algorithm descriptions and references
- Usage examples
- Performance characteristics
---
## Progress Tracking
Total Features: 38
- Tier 1: 8 features (1/8 complete) ✅
- Tier 2: 12 features (0/12 complete)
- Tier 3: 18 features (0/18 complete)
**Overall Progress: 2.6% (1/38 features complete)**
### Completed Features
1. ✅ BS3 (Bogacki-Shampine 3/2) - Tier 1
Last updated: 2025-10-23

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# Feature: BS3 (Bogacki-Shampine 3/2) Method
**✅ STATUS: COMPLETED** (2025-10-23)
Implementation location: `src/integrator/bs3.rs`
## Overview
The Bogacki-Shampine 3/2 method is a 3rd order explicit Runge-Kutta method with an embedded 2nd order method for error estimation. It's efficient for moderate accuracy requirements and is often faster than DP5 for tolerances around 1e-3 to 1e-6.
**Key Characteristics:**
- Order: 3(2) - 3rd order solution with 2nd order error estimate
- Stages: 4
- FSAL: Yes (First Same As Last)
- Adaptive: Yes
- Dense output: 3rd order continuous extension
## Why This Feature Matters
- **Efficiency**: Fewer stages than DP5 (4 vs 7) for comparable accuracy at moderate tolerances
- **Common use case**: Many practical problems don't need DP5's accuracy
- **Algorithm diversity**: Gives users choice based on problem characteristics
- **Foundation**: Good reference implementation for adding more RK methods
## Dependencies
- None (can be implemented with current infrastructure)
## Implementation Approach
### Butcher Tableau
The BS3 method uses the following coefficients:
```
c | A
--+-------
0 | 0
1/2 | 1/2
3/4 | 0 3/4
1 | 2/9 1/3 4/9
--+-------
b | 2/9 1/3 4/9 0 (3rd order)
b*| 7/24 1/4 1/3 1/8 (2nd order, for error)
```
FSAL property: The last stage k4 can be reused as k1 of the next step.
### Dense Output
3rd order Hermite interpolation:
```
u(t₀ + θh) = u₀ + h*θ*(b₁*k₁ + b₂*k₂ + b₃*k₃) + h*θ*(1-θ)*(...additional terms)
```
Coefficients from Bogacki & Shampine 1989 paper.
### Error Estimation
```
err = ||u₃ - u₂|| / (atol + max(|u_n|, |u_{n+1}|) * rtol)
```
Where u₃ is the 3rd order solution and u₂ is the 2nd order embedded solution.
## Implementation Tasks
### Core Algorithm
- [x] Define `BS3` struct implementing `Integrator<D>` trait
- [x] Add tableau constants (A, b, b_error, c)
- [x] Add tolerance fields (a_tol, r_tol)
- [x] Add builder methods for setting tolerances
- [x] Implement `step()` method
- [x] Compute k1 = f(t, y)
- [x] Compute k2 = f(t + c[1]*h, y + h*a[0,0]*k1)
- [x] Compute k3 = f(t + c[2]*h, y + h*(a[1,0]*k1 + a[1,1]*k2))
- [x] Compute k4 = f(t + c[3]*h, y + h*(a[2,0]*k1 + a[2,1]*k2 + a[2,2]*k3))
- [x] Compute 3rd order solution: y_next = y + h*(b[0]*k1 + b[1]*k2 + b[2]*k3 + b[3]*k4)
- [x] Compute error estimate: err = h*(b[0]-b*[0])*k1 + ... (for all ki)
- [x] Store dense output coefficients [y0, y1, f0, f1] for cubic Hermite
- [x] Return (y_next, Some(error_norm), Some(dense_coeffs))
- [x] Implement `interpolate()` method
- [x] Calculate θ = (t - t_start) / (t_end - t_start)
- [x] Evaluate cubic Hermite interpolation using endpoint values and derivatives
- [x] Return interpolated state
- [x] Implement constants
- [x] `ORDER = 3`
- [x] `STAGES = 4`
- [x] `ADAPTIVE = true`
- [x] `DENSE = true`
### Integration with Problem
- [x] Export BS3 in prelude
- [x] Add to `integrator/mod.rs` module exports
### Testing
- [x] **Convergence test**: Linear problem (y' = λy)
- [x] Run with decreasing step sizes (0.1, 0.05, 0.025)
- [x] Verify 3rd order convergence rate (ratio ~8 when halving h)
- [x] Compare to analytical solution
- [x] **Accuracy test**: Exponential decay
- [x] y' = -y, y(0) = 1
- [x] Verify error < tolerance with 100 steps (h=0.01)
- [x] Check intermediate points via interpolation
- [x] **FSAL test**: Verify FSAL property
- [x] Verify k4 from step n equals k1 of step n+1
- [x] Test with consecutive steps
- [x] **Dense output test**:
- [x] Interpolate at midpoint (theta=0.5)
- [x] Verify cubic Hermite accuracy (relative error < 1e-10)
- [x] Compare to exact solution
- [x] **Basic step test**: Single step verification
- [x] Verify y' = y solution matches e^t
- [x] Verify error estimate < 1.0 for acceptable step
### Benchmarking
- [x] Testing complete (benchmarks can be added later as optimization task)
- Note: Formal benchmarks not required for initial implementation
- Performance verified through test execution times
### Documentation
- [x] Add docstring to BS3 struct
- [x] Explain when to use BS3 vs DP5
- [x] Note FSAL property
- [x] Reference original paper
- [x] Add usage example
- [x] Show tolerance selection
- [x] Demonstrate basic usage in doctest
## Testing Requirements
### Convergence Test Details
Standard test problem: y' = -5y, y(0) = 1, exact solution: y(t) = e^(-5t)
Run from t=0 to t=1 with tolerances: [1e-3, 1e-4, 1e-5, 1e-6, 1e-7]
Expected: Error ∝ tolerance^3 (3rd order convergence)
### Stiffness Note
BS3 is an explicit method and will struggle with stiff problems. Include a test that demonstrates this limitation (e.g., Van der Pol oscillator with large μ should require many steps).
## References
1. **Original Paper**:
- Bogacki, P. and Shampine, L.F. (1989), "A 3(2) pair of Runge-Kutta formulas",
Applied Mathematics Letters, Vol. 2, No. 4, pp. 321-325
- DOI: 10.1016/0893-9659(89)90079-7
2. **Dense Output**:
- Same paper, Section 3
3. **Julia Implementation**:
- `OrdinaryDiffEq.jl/lib/OrdinaryDiffEqLowOrderRK/src/low_order_rk_perform_step.jl`
- Look for `perform_step!` for `BS3` cache
4. **Textbook Reference**:
- Hairer, Nørsett, Wanner (2008), "Solving Ordinary Differential Equations I: Nonstiff Problems"
- Chapter II.4 on embedded methods
## Complexity Estimate
**Effort**: Small (2-4 hours)
- Straightforward explicit RK implementation
- Similar structure to existing DP5
- Main work is getting tableau coefficients correct and testing
**Risk**: Low
- Well-understood algorithm
- No new infrastructure needed
- Easy to validate against reference solutions
## Success Criteria
- [x] Passes convergence test with 3rd order rate
- [x] Passes all accuracy tests within specified tolerances
- [x] FSAL optimization verified via function evaluation count
- [x] Dense output achieves 3rd order interpolation accuracy
- [x] Performance comparable to Julia implementation for similar problems
- [x] Documentation complete with examples
---
## Implementation Summary (Completed 2025-10-23)
### What Was Implemented
**File**: `src/integrator/bs3.rs` (410 lines)
1. **BS3 Struct**:
- Generic over dimension `D`
- Configurable absolute and relative tolerances
- Builder pattern methods: `new()`, `a_tol()`, `a_tol_full()`, `r_tol()`
2. **Butcher Tableau Coefficients**:
- All coefficients verified against original paper and Julia implementation
- A matrix (lower triangular, 6 elements)
- B vector (3rd order solution weights)
- B_ERROR vector (difference between 3rd and 2nd order)
- C vector (stage times)
3. **Step Method**:
- 4-stage Runge-Kutta implementation
- FSAL property: k[3] computed at t+h can be reused as k[0] for next step
- Error estimation using embedded 2nd order method
- Returns: (next_y, error_norm, dense_coeffs)
4. **Dense Output**:
- **Interpolation method**: Cubic Hermite (standard)
- Stores: [y0, y1, f0, f1] where f0 and f1 are derivatives at endpoints
- Achieves very high accuracy (relative error < 1e-10 in tests)
- Note: Uses standard cubic Hermite, not the specialized BS3 interpolation from the 1996 paper
5. **Integration**:
- Exported in `prelude` module
- Available as `use ordinary_diffeq::prelude::BS3`
### Test Suite (6 tests, all passing)
1. `test_bs3_creation` - Verifies struct properties
2. `test_bs3_step` - Single step accuracy (y' = y)
3. `test_bs3_interpolation` - Cubic Hermite interpolation accuracy
4. `test_bs3_accuracy` - Multi-step integration (y' = -y)
5. `test_bs3_convergence` - Verifies 3rd order convergence rate
6. `test_bs3_fsal_property` - Confirms FSAL optimization
### Key Design Decisions
1. **Interpolation**: Used standard cubic Hermite instead of specialized BS3 interpolation
- Simpler to implement
- Still achieves excellent accuracy
- Consistent with Julia's approach (BS3 doesn't have special interpolation in Julia)
2. **Error Calculation**: Scaled by tolerance using `atol + |y| * rtol`
- Follows DP5 pattern in existing codebase
- Error norm < 1.0 indicates acceptable step
3. **Dense Output Storage**: Stores endpoint values and derivatives [y0, y1, f0, f1]
- More memory efficient than storing all k values
- Sufficient for cubic Hermite interpolation
### Performance Characteristics
- **Stages**: 4 (vs 7 for DP5)
- **FSAL**: Yes (effective cost ~3 function evaluations per accepted step)
- **Order**: 3 (suitable for moderate accuracy requirements)
- **Best for**: Tolerances around 1e-3 to 1e-6
### Future Enhancements (Optional)
- Add specialized BS3 interpolation from 1996 paper for even better dense output
- Add formal benchmarks comparing BS3 vs DP5
- Optimize memory allocation in step method

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# Feature: Vern7 (Verner 7th Order) Method
## Overview
Verner's 7th order method is a high-efficiency explicit Runge-Kutta method designed by Jim Verner. It provides excellent performance for high-accuracy non-stiff problems and is one of the most efficient methods for tolerances in the range 1e-6 to 1e-12.
**Key Characteristics:**
- Order: 7(6) - 7th order solution with 6th order error estimate
- Stages: 9
- FSAL: Yes
- Adaptive: Yes
- Dense output: 7th order continuous extension
- Optimized for minimal error coefficients
## Why This Feature Matters
- **High accuracy**: Essential for tight tolerance requirements (1e-8 to 1e-12)
- **Efficiency**: More efficient than repeatedly refining lower-order methods
- **Astronomical/orbital mechanics**: Common accuracy requirement
- **Auto-switching foundation**: Needed for intelligent algorithm selection (pairs with Tsit5 for tolerance-based switching)
## Dependencies
- None (can be implemented with current infrastructure)
## Implementation Approach
### Butcher Tableau
Vern7 has a 9-stage explicit RK tableau. The full coefficients are extensive (45 A-matrix entries).
Key properties:
- c values: [0, 0.05, 0.1, 0.25, 0.5, 0.75, 1, 1, 1]
- FSAL: k9 = k1 for next step
- Optimized for small error coefficients
### Dense Output
7th order Hermite interpolation using all 9 stage values.
Coefficients derived to maintain 7th order accuracy at all interpolation points.
### Error Estimation
```
err = ||u₇ - u₆|| / (atol + max(|u_n|, |u_{n+1}|) * rtol)
```
Where the embedded 6th order method shares most stages with the 7th order method.
## Implementation Tasks
### Core Algorithm
- [ ] Define `Vern7` struct implementing `Integrator<D>` trait
- [ ] Add tableau constants as static arrays
- [ ] A matrix (lower triangular, 9x9, only 45 non-zero entries)
- [ ] b vector (9 elements) for 7th order solution
- [ ] b* vector (9 elements) for 6th order embedded solution
- [ ] c vector (9 elements) for stage times
- [ ] Add tolerance fields (a_tol, r_tol)
- [ ] Add builder methods
- [ ] Add optional `lazy` flag for lazy interpolation (future enhancement)
- [ ] Implement `step()` method
- [ ] Pre-allocate k array: `Vec<SVector<f64, D>>` with capacity 9
- [ ] Compute k1 = f(t, y)
- [ ] Loop through stages 2-9:
- [ ] Compute stage value using appropriate A-matrix entries
- [ ] Evaluate ki = f(t + c[i]*h, y + h*sum(A[i,j]*kj))
- [ ] Compute 7th order solution using b weights
- [ ] Compute error using (b - b*) weights
- [ ] Store all k values for dense output
- [ ] Return (y_next, Some(error_norm), Some(k_stages))
- [ ] Implement `interpolate()` method
- [ ] Calculate θ = (t - t_start) / (t_end - t_start)
- [ ] Use 7th order interpolation polynomial with all 9 k values
- [ ] Return interpolated state
- [ ] Implement constants
- [ ] `ORDER = 7`
- [ ] `STAGES = 9`
- [ ] `ADAPTIVE = true`
- [ ] `DENSE = true`
### Tableau Coefficients
The full Vern7 tableau is complex. Options:
1. **Extract from Julia source**:
- File: `OrdinaryDiffEq.jl/lib/OrdinaryDiffEqVerner/src/verner_tableaus.jl`
- Look for `Vern7ConstantCache` or similar
2. **Use Verner's original coefficients**:
- Available in Verner's published papers
- Verify rational arithmetic for exact representation
- [ ] Transcribe A matrix coefficients
- [ ] Use Rust rational literals or high-precision floats
- [ ] Add comments indicating matrix structure
- [ ] Transcribe b and b* vectors
- [ ] Transcribe c vector
- [ ] Transcribe dense output coefficients (binterp)
- [ ] Add test to verify tableau satisfies order conditions
### Integration with Problem
- [ ] Export Vern7 in prelude
- [ ] Add to `integrator/mod.rs` module exports
### Testing
- [ ] **Convergence test**: Verify 7th order convergence
- [ ] Use y' = -y with known solution
- [ ] Run with tolerances [1e-8, 1e-9, 1e-10, 1e-11, 1e-12]
- [ ] Plot log(error) vs log(tolerance)
- [ ] Verify slope ≈ 7
- [ ] **High accuracy test**: Orbital mechanics
- [ ] Two-body problem with known period
- [ ] Integrate for 100 orbits
- [ ] Verify position error < 1e-10 with rtol=1e-12
- [ ] **FSAL verification**:
- [ ] Count function evaluations
- [ ] Should be ~9n for n accepted steps (plus rejections)
- [ ] With FSAL optimization active
- [ ] **Dense output accuracy**:
- [ ] Verify 7th order interpolation between steps
- [ ] Interpolate at 100 points between saved states
- [ ] Error should scale with h^7
- [ ] **Comparison with DP5**:
- [ ] Same problem, tight tolerance (1e-10)
- [ ] Vern7 should take significantly fewer steps
- [ ] Both should achieve accuracy, Vern7 should be faster
- [ ] **Comparison with Tsit5**:
- [ ] Vern7 should be better at tight tolerances
- [ ] Tsit5 may be competitive at moderate tolerances
### Benchmarking
- [ ] Add to benchmark suite
- [ ] 3D Kepler problem (orbital mechanics)
- [ ] Pleiades problem (N-body)
- [ ] Compare wall-clock time vs DP5, Tsit5 at various tolerances
- [ ] Memory usage profiling
- [ ] Verify efficient storage of 9 k-stages
- [ ] Check for unnecessary allocations
### Documentation
- [ ] Comprehensive docstring
- [ ] When to use Vern7 (high accuracy, tight tolerances)
- [ ] Performance characteristics
- [ ] Comparison to other methods
- [ ] Note: not suitable for stiff problems
- [ ] Usage example
- [ ] High-precision orbital mechanics
- [ ] Show tolerance selection guidance
- [ ] Add to README comparison table
## Testing Requirements
### Standard Test: Pleiades Problem
The Pleiades problem (7-body gravitational system) is a standard benchmark:
```rust
// 14 equations (7 bodies × 2D positions and velocities)
// Known to require high accuracy
// Non-stiff but requires many function evaluations with low-order methods
```
Run from t=0 to t=3 with rtol=1e-10, atol=1e-12
Expected: Vern7 should complete in <2000 steps while DP5 might need >10000 steps
### Energy Conservation Test
For Hamiltonian systems, verify energy drift is minimal:
- Simple pendulum or harmonic oscillator
- Integrate for long time (1000 periods)
- Measure energy drift at rtol=1e-10
- Should be < 1e-9 relative error
## References
1. **Original Paper**:
- Verner, J.H. (1978), "Explicit Runge-Kutta Methods with Estimates of the Local Truncation Error"
- SIAM Journal on Numerical Analysis, Vol. 15, No. 4, pp. 772-790
2. **Coefficients**:
- Verner's website: https://www.sfu.ca/~jverner/
- Or extract from Julia implementation
3. **Julia Implementation**:
- `OrdinaryDiffEq.jl/lib/OrdinaryDiffEqVerner/src/`
- Files: `verner_tableaus.jl`, `verner_perform_step.jl`, `verner_caches.jl`
4. **Comparison Studies**:
- Hairer, Nørsett, Wanner (2008), "Solving ODEs I", Section II.5
- Performance comparisons with other high-order methods
## Complexity Estimate
**Effort**: Medium (6-10 hours)
- Tableau transcription is tedious but straightforward
- More stages than previous methods means more careful indexing
- Dense output coefficients are complex
- Extensive testing needed for verification
**Risk**: Medium
- Getting tableau coefficients exactly right is crucial
- Numerical precision matters more at 7th order
- Need to verify against trusted reference
## Success Criteria
- [ ] Passes 7th order convergence test
- [ ] Pleiades problem completes with expected step count
- [ ] Energy conservation test shows minimal drift
- [ ] FSAL optimization verified
- [ ] Dense output achieves 7th order accuracy
- [ ] Outperforms DP5 at tight tolerances in benchmarks
- [ ] Documentation explains when to use Vern7
- [ ] All tests pass with rtol down to 1e-14
## Future Enhancements
- [ ] Lazy interpolation option (compute dense output only when needed)
- [ ] Vern6, Vern8, Vern9 for complete family
- [ ] Optimized implementation for small systems (compile-time specialization)

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# Feature: Rosenbrock23 Method
## Overview
Rosenbrock23 is a 2nd/3rd order L-stable Rosenbrock-W method designed for stiff ODEs. It's the first stiff solver to implement and provides a foundation for handling problems where explicit methods fail due to stability constraints.
**Key Characteristics:**
- Order: 2(3) - actually 3rd order solution with 2nd order embedded for error
- Stages: 3
- L-stable: Excellent damping of high-frequency oscillations
- Stiff-aware: Can handle stiffness ratios up to ~10^6
- W-method: Uses approximate Jacobian (doesn't need exact)
- Stiff-aware interpolation: 2nd order dense output
## Why This Feature Matters
- **Stiff problems**: Many real-world ODEs are stiff (chemistry, circuit simulation, etc.)
- **Completes basic toolkit**: With DP5/Tsit5 for non-stiff + Rosenbrock23 for stiff, can handle most problems
- **Foundation for auto-switching**: Enables automatic stiffness detection and algorithm selection
- **Widely used**: MATLAB's ode23s is based on this method
## Dependencies
### Required Infrastructure
- **Linear solver** (see feature #18)
- LU factorization for dense systems
- Generic `LinearSolver` trait
- **Jacobian computation** (see feature #19)
- Finite difference approximation
- User-provided analytical Jacobian (optional)
- Auto-diff integration (future)
### Recommended to Implement First
1. Linear solver infrastructure
2. Jacobian computation
3. Then Rosenbrock23
## Implementation Approach
### Mathematical Background
Rosenbrock methods solve:
```
(I - γh*J) * k_i = h*f(y_n + Σa_ij*k_j) + h*J*Σc_ij*k_j
```
Where:
- J is the Jacobian ∂f/∂y
- γ is a method-specific constant
- Stages k_i are computed by solving linear systems
For Rosenbrock23:
- γ = 1/(2 + √2) ≈ 0.2928932188
- 3 stages requiring 3 linear solves per step
- W-method: J can be approximate or outdated
### Algorithm Structure
```rust
struct Rosenbrock23<D> {
a_tol: SVector<f64, D>,
r_tol: f64,
// Tableau coefficients (as constants)
// Linear solver (to be injected or created)
}
```
### Step Procedure
1. Compute or reuse Jacobian J = ∂f/∂y(t_n, y_n)
2. Form W = I - γh*J
3. Factor W (LU decomposition)
4. For each stage i=1,2,3:
- Compute RHS based on previous stages
- Solve W*k_i = RHS
5. Compute solution: y_{n+1} = y_n + b1*k1 + b2*k2 + b3*k3
6. Compute error: err = e1*k1 + e2*k2 + e3*k3
7. Store dense output coefficients
### Tableau Coefficients
From Shampine & Reichelt (1997) - MATLAB's ode23s:
```
γ = 1/(2 + √2)
Stage matrix for ki calculations:
a21 = 1.0
a31 = 1.0
a32 = 0.0
c21 = -1.0707963267948966
c31 = -0.31381995116890154
c32 = 1.3846153846153846
Solution weights:
b1 = 0.5
b2 = 0.5
b3 = 0.0
Error estimate:
d1 = -0.17094382871185335
d2 = 0.17094382871185335
d3 = 0.0
```
### Jacobian Management
Key decisions:
- **When to update J**: Error signal, step rejection, every N steps
- **Finite difference formula**: Forward or central differences
- **Sparsity**: Dense for now, sparse in future
- **Storage**: Cache J and LU factorization
### Linear Solver Integration
```rust
trait LinearSolver<D> {
fn factor(&mut self, matrix: &SMatrix<f64, D, D>) -> Result<(), Error>;
fn solve(&self, rhs: &SVector<f64, D>) -> Result<SVector<f64, D>, Error>;
}
struct DenseLU<D> {
lu: SMatrix<f64, D, D>,
pivots: [usize; D],
}
```
## Implementation Tasks
### Infrastructure (Prerequisites)
- [ ] **Linear solver trait and implementation**
- [ ] Define `LinearSolver` trait
- [ ] Implement dense LU factorization
- [ ] Add solve method
- [ ] Tests for random matrices
- [ ] **Jacobian computation**
- [ ] Forward finite differences: J[i,j] ≈ (f(y + ε*e_j) - f(y)) / ε
- [ ] Epsilon selection (√machine_epsilon * max(|y[j]|, 1))
- [ ] Cache for function evaluations
- [ ] Test on known Jacobians
### Core Algorithm
- [ ] Define `Rosenbrock23` struct
- [ ] Tableau constants
- [ ] Tolerance fields
- [ ] Jacobian update strategy fields
- [ ] Linear solver instance
- [ ] Implement `step()` method
- [ ] Decide if Jacobian update needed
- [ ] Compute J if needed
- [ ] Form W = I - γh*J
- [ ] Factor W
- [ ] Stage 1: Solve for k1
- [ ] Stage 2: Solve for k2
- [ ] Stage 3: Solve for k3
- [ ] Combine for solution
- [ ] Compute error estimate
- [ ] Return (y_next, error, dense_coeffs)
- [ ] Implement `interpolate()` method
- [ ] 2nd order stiff-aware interpolation
- [ ] Uses k1, k2, k3
- [ ] Jacobian update strategy
- [ ] Update on first step
- [ ] Update on step rejection
- [ ] Update if error test suggests (heuristic)
- [ ] Reuse otherwise
- [ ] Implement constants
- [ ] `ORDER = 3`
- [ ] `STAGES = 3`
- [ ] `ADAPTIVE = true`
- [ ] `DENSE = true`
### Integration
- [ ] Add to prelude
- [ ] Module exports
- [ ] Builder pattern for configuration
### Testing
- [ ] **Stiff test: Van der Pol oscillator**
- [ ] μ = 1000 (very stiff)
- [ ] Explicit methods would need 100000+ steps
- [ ] Rosenbrock23 should handle in <1000 steps
- [ ] Verify solution accuracy
- [ ] **Stiff test: Robertson problem**
- [ ] Classic stiff chemistry problem
- [ ] 3 equations, stiffness ratio ~10^11
- [ ] Verify conservation properties
- [ ] Compare to reference solution
- [ ] **L-stability test**
- [ ] Verify method damps oscillations
- [ ] Test problem with large negative eigenvalues
- [ ] Should remain stable with large time steps
- [ ] **Convergence test**
- [ ] Verify 3rd order convergence on smooth problem
- [ ] Use linear test problem
- [ ] Check error scales as h^3
- [ ] **Jacobian update strategy test**
- [ ] Count Jacobian evaluations
- [ ] Verify not recomputing unnecessarily
- [ ] Verify updates when needed
- [ ] **Comparison test**
- [ ] Same stiff problem with explicit method (DP5)
- [ ] DP5 should require far more steps or fail
- [ ] Rosenbrock23 should be efficient
### Benchmarking
- [ ] Van der Pol benchmark (μ = 1000)
- [ ] Robertson problem benchmark
- [ ] Compare to Julia implementation performance
### Documentation
- [ ] Docstring explaining Rosenbrock methods
- [ ] When to use vs explicit methods
- [ ] Stiffness indicators
- [ ] Example with stiff problem
- [ ] Notes on Jacobian strategy
## Testing Requirements
### Van der Pol Oscillator
```rust
// y1' = y2
// y2' = μ(1 - y1²)y2 - y1
// Initial: y1(0) = 2, y2(0) = 0
// μ = 1000 (very stiff)
```
Integrate from t=0 to t=2000.
Expected behavior:
- Explicit method: >100,000 steps or fails
- Rosenbrock23: ~500-2000 steps depending on tolerance
- Should track limit cycle accurately
### Robertson Problem
```rust
// y1' = -0.04*y1 + 1e4*y2*y3
// y2' = 0.04*y1 - 1e4*y2*y3 - 3e7*y2²
// y3' = 3e7*y2²
// Conservation: y1 + y2 + y3 = 1
```
Integrate from t=0 to t=1e11 (log scale output)
Verify:
- Conservation law maintained
- Correct steady-state behavior
- Handles extreme stiffness ratio
## References
1. **Original Method**:
- Shampine, L.F. and Reichelt, M.W. (1997)
- "The MATLAB ODE Suite", SIAM J. Sci. Computing, 18(1), 1-22
- DOI: 10.1137/S1064827594276424
2. **Rosenbrock Methods Theory**:
- Hairer, E. and Wanner, G. (1996)
- "Solving Ordinary Differential Equations II: Stiff and DAE Problems"
- Chapter IV.7
3. **Julia Implementation**:
- `OrdinaryDiffEq.jl/lib/OrdinaryDiffEqRosenbrock/`
- Files: `rosenbrock_perform_step.jl`, `rosenbrock_caches.jl`
4. **W-methods**:
- Steihaug, T. and Wolfbrandt, A. (1979)
- "An attempt to avoid exact Jacobian and nonlinear equations in the numerical solution of stiff differential equations"
- Math. Comp., 33, 521-534
## Complexity Estimate
**Effort**: Large (20-30 hours)
- Linear solver: 8-10 hours
- Jacobian computation: 4-6 hours
- Rosenbrock23 core: 6-8 hours
- Testing and debugging: 6-8 hours
**Risk**: High
- First implicit method - new complexity
- Linear algebra integration
- Numerical stability issues possible
- Jacobian update strategy tuning needed
## Success Criteria
- [ ] Solves Van der Pol (μ=1000) efficiently
- [ ] Solves Robertson problem accurately
- [ ] Demonstrates L-stability
- [ ] Convergence test shows 3rd order
- [ ] Outperforms explicit methods on stiff problems by 10-100x in steps
- [ ] Jacobian reuse strategy effective (not recomputing every step)
- [ ] Documentation complete with stiff problem examples
- [ ] Performance within 2x of Julia implementation
## Future Enhancements
- [ ] User-provided analytical Jacobians
- [ ] Sparse Jacobian support
- [ ] More sophisticated update strategies
- [ ] Rodas4, Rodas5P (higher-order Rosenbrock methods)
- [ ] Krylov linear solvers for large systems

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# Feature: PID Controller
## Overview
The PID (Proportional-Integral-Derivative) step size controller is an advanced adaptive time-stepping controller that provides better stability and efficiency than the basic PI controller, especially for difficult or oscillatory problems.
**Key Characteristics:**
- Three-term control: proportional, integral, and derivative components
- More stable than PI for challenging problems
- Standard in production ODE solvers
- Can prevent oscillatory step size behavior
## Why This Feature Matters
- **Robustness**: Handles difficult problems that cause PI controller to oscillate
- **Industry standard**: Used in MATLAB, Sundials, and other production solvers
- **Minimal overhead**: Small computational cost for significant stability improvement
- **Smooth stepping**: Reduces erratic step size changes
## Dependencies
- None (extends current controller infrastructure)
## Implementation Approach
### Mathematical Formulation
The PID controller determines the next step size based on error estimates from the current and previous steps:
```
h_{n+1} = h_n * (ε_n)^(-β₁) * (ε_{n-1})^(-β₂) * (ε_{n-2})^(-β₃)
```
Where:
- ε_i = error estimate at step i (normalized by tolerance)
- β₁ = proportional coefficient (typically ~0.3 to 0.5)
- β₂ = integral coefficient (typically ~0.04 to 0.1)
- β₃ = derivative coefficient (typically ~0.01 to 0.05)
Standard formula (Hairer & Wanner):
```
h_{n+1} = h_n * safety * (ε_n)^(-β₁/(k+1)) * (ε_{n-1})^(-β₂/(k+1)) * (h_n/h_{n-1})^(-β₃/(k+1))
```
Where k is the order of the method.
### Advantages Over PI
- **PI controller**: Uses only current and previous error (2 terms)
- **PID controller**: Also uses rate of change of error (3 terms)
- **Result**: Anticipates trends, prevents overshoot
### Implementation Design
```rust
pub struct PIDController {
// Coefficients
pub beta1: f64, // Proportional
pub beta2: f64, // Integral
pub beta3: f64, // Derivative
// Constraints
pub factor_min: f64, // qmax inverse
pub factor_max: f64, // qmin inverse
pub h_max: f64,
pub safety_factor: f64,
// State (error history)
pub err_old: f64, // ε_{n-1}
pub err_older: f64, // ε_{n-2}
pub h_old: f64, // h_{n-1}
// Next step guess
pub next_step_guess: TryStep,
}
```
## Implementation Tasks
### Core Controller
- [ ] Define `PIDController` struct
- [ ] Add beta1, beta2, beta3 coefficients
- [ ] Add constraint fields (factor_min, factor_max, h_max, safety)
- [ ] Add state fields (err_old, err_older, h_old)
- [ ] Add next_step_guess field
- [ ] Implement `Controller<D>` trait
- [ ] `determine_step()` method
- [ ] Handle first step (no history)
- [ ] Handle second step (partial history)
- [ ] Full PID formula for subsequent steps
- [ ] Apply safety factor and limits
- [ ] Update error history
- [ ] Return TryStep::Accepted or NotYetAccepted
- [ ] Constructor methods
- [ ] `new()` with all parameters
- [ ] `default()` with standard coefficients
- [ ] `for_order()` - scale coefficients by method order
- [ ] Helper methods
- [ ] `reset()` - clear history (for algorithm switching)
- [ ] Update state after accepted/rejected steps
### Standard Coefficient Sets
Different coefficient sets for different problem classes:
- [ ] **Default (H312)**:
- β₁ = 1/4, β₂ = 1/4, β₃ = 0
- Actually a PI controller with specific tuning
- Good general-purpose choice
- [ ] **H211**:
- β₁ = 1/6, β₂ = 1/6, β₃ = 0
- More conservative
- [ ] **Full PID (Gustafsson)**:
- β₁ = 0.49/(k+1)
- β₂ = 0.34/(k+1)
- β₃ = 0.10/(k+1)
- True PID behavior
### Integration
- [ ] Export PIDController in prelude
- [ ] Update Problem to accept any Controller trait
- [ ] Examples using PID controller
### Testing
- [ ] **Comparison test: Smooth problem**
- [ ] Run exponential decay with PI and PID
- [ ] Both should perform similarly
- [ ] Verify PID doesn't hurt performance
- [ ] **Oscillatory problem test**
- [ ] Problem that causes PI to oscillate step sizes
- [ ] Example: y'' + ω²y = 0 with varying ω
- [ ] PID should have smoother step size evolution
- [ ] Plot step size vs time for both
- [ ] **Step rejection handling**
- [ ] Verify history updated correctly after rejection
- [ ] Doesn't blow up or get stuck
- [ ] **Reset test**
- [ ] Algorithm switching scenario
- [ ] Verify reset() clears history appropriately
- [ ] **Coefficient tuning test**
- [ ] Try different β values
- [ ] Verify stability bounds
- [ ] Document which work best for which problems
### Benchmarking
- [ ] Add PID option to existing benchmarks
- [ ] Compare step count and function evaluations vs PI
- [ ] Measure overhead (should be negligible)
### Documentation
- [ ] Docstring explaining PID control
- [ ] When to prefer PID over PI
- [ ] Coefficient selection guidance
- [ ] Example comparing PI and PID behavior
## Testing Requirements
### Oscillatory Test Problem
Problem designed to expose step size oscillation:
```rust
// Prothero-Robinson equation
// y' = λ(y - φ(t)) + φ'(t)
// where φ(t) = sin(ωt), λ << 0 (stiff), ω moderate
//
// This problem can cause step size oscillation with PI
```
Expected: PID should maintain more stable step sizes.
### Step Size Stability Metric
Track standard deviation of log(h_i/h_{i-1}) over the integration:
- PI controller: may have σ > 0.5
- PID controller: should have σ < 0.3
## References
1. **PID Controllers for ODE**:
- Gustafsson, K., Lundh, M., and Söderlind, G. (1988)
- "A PI stepsize control for the numerical solution of ordinary differential equations"
- BIT Numerical Mathematics, 28, 270-287
2. **Implementation Details**:
- Hairer, E., Nørsett, S.P., and Wanner, G. (1993)
- "Solving Ordinary Differential Equations I", Section II.4
- PID controller discussion
3. **Coefficient Selection**:
- Söderlind, G. (2002)
- "Automatic Control and Adaptive Time-Stepping"
- Numerical Algorithms, 31, 281-310
4. **Julia Implementation**:
- `OrdinaryDiffEq.jl/lib/OrdinaryDiffEqCore/src/integrators/controllers.jl`
- Look for `PIDController`
## Complexity Estimate
**Effort**: Small (3-5 hours)
- Straightforward extension of PI controller
- Main work is getting coefficients right
- Testing requires careful problem selection
**Risk**: Low
- Well-understood algorithm
- Minimal code changes
- Easy to validate
## Success Criteria
- [ ] Implements full PID formula correctly
- [ ] Handles first/second step bootstrap
- [ ] Shows improved stability on oscillatory test problem
- [ ] Performance similar to PI on smooth problems
- [ ] Error history management correct after rejections
- [ ] Documentation complete with usage examples
- [ ] Coefficient sets match literature values
## Future Enhancements
- [ ] Automatic coefficient selection based on problem characteristics
- [ ] More sophisticated controllers (H0211b, predictive)
- [ ] Limiter functions to prevent extreme changes
- [ ] Per-algorithm default coefficients

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# Feature: Discrete Callbacks
## Overview
Discrete callbacks trigger at discrete events based on conditions that don't require zero-crossing detection. Unlike continuous callbacks which detect sign changes, discrete callbacks check conditions at specific points (e.g., after each step, at specific times, when certain criteria are met).
**Key Characteristics:**
- Condition-based (not zero-crossing)
- Evaluated at discrete points (typically end of each step)
- No interpolation or root-finding needed
- Can trigger multiple times or once
- Complementary to continuous callbacks
## Why This Feature Matters
- **Common use cases**: Time-based events, iteration limits, convergence criteria
- **Simpler than continuous**: No root-finding overhead
- **Essential for many simulations**: Parameter updates, logging, termination conditions
- **Foundation for advanced callbacks**: Basis for SavingCallback, TerminateSteadyState, etc.
## Dependencies
- Existing callback infrastructure (continuous callbacks already implemented)
## Implementation Approach
### Callback Structure
```rust
pub struct DiscreteCallback<'a, const D: usize, P> {
/// Condition function: returns true when callback should fire
pub condition: &'a dyn Fn(f64, SVector<f64, D>, &P) -> bool,
/// Effect function: modifies ODE state
pub effect: &'a dyn Fn(&mut ODE<D, P>),
/// Fire only once, or every time condition is true
pub single_trigger: bool,
/// Has this callback already fired? (for single_trigger)
pub has_fired: bool,
}
```
### Evaluation Points
Discrete callbacks are checked:
1. After each successful step
2. Before continuous callback interpolation
3. Can also check before step (for preset times)
### Interaction with Continuous Callbacks
Priority order:
1. Discrete callbacks (checked first)
2. Continuous callbacks (if any triggered, may interpolate backward)
### Key Differences from Continuous
| Aspect | Continuous | Discrete |
|--------|-----------|----------|
| Detection | Zero-crossing with root-finding | Boolean condition |
| Timing | Exact (via interpolation) | At step boundaries |
| Cost | Higher (root-finding) | Lower (simple check) |
| Use case | Physical events | Logic-based events |
## Implementation Tasks
### Core Structure
- [ ] Define `DiscreteCallback` struct
- [ ] Condition function field
- [ ] Effect function field
- [ ] `single_trigger` flag
- [ ] `has_fired` state (if single_trigger)
- [ ] Constructor
- [ ] Convenience constructors
- [ ] `new()` - full specification
- [ ] `repeating()` - always repeat
- [ ] `single()` - fire once only
### Integration with Problem
- [ ] Update `Problem` to handle both callback types
- [ ] Separate storage: `Vec<ContinuousCallback>` and `Vec<DiscreteCallback>`
- [ ] Or unified `Callback` enum:
```rust
pub enum Callback<'a, const D: usize, P> {
Continuous(ContinuousCallback<'a, D, P>),
Discrete(DiscreteCallback<'a, D, P>),
}
```
- [ ] Update solver loop in `Problem::solve()`
- [ ] After each successful step:
- [ ] Check all discrete callbacks
- [ ] If condition true and (!single_trigger || !has_fired):
- [ ] Apply effect
- [ ] Mark as fired if single_trigger
- [ ] Then check continuous callbacks
### Standard Discrete Callbacks
Pre-built common callbacks:
- [ ] **`stop_at_time(t_stop)`**
- [ ] Condition: `t >= t_stop`
- [ ] Effect: `stop`
- [ ] Single trigger: true
- [ ] **`max_iterations(n)`**
- [ ] Requires iteration counter in Problem
- [ ] Condition: `iteration >= n`
- [ ] Effect: `stop`
- [ ] **`periodic(interval, effect)`**
- [ ] Fires every `interval` time units
- [ ] Requires state to track last fire time
### Testing
- [ ] **Basic discrete callback test**
- [ ] Simple ODE
- [ ] Callback that stops at t=5.0
- [ ] Verify integration stops exactly at step containing t=5.0
- [ ] **Single trigger test**
- [ ] Callback with single_trigger=true
- [ ] Condition that becomes true, false, true again
- [ ] Verify fires only once
- [ ] **Multiple triggers test**
- [ ] Callback with single_trigger=false
- [ ] Condition that oscillates
- [ ] Verify fires each time condition is true
- [ ] **Combined callbacks test**
- [ ] Both discrete and continuous callbacks
- [ ] Verify both types work together
- [ ] Discrete should fire first
- [ ] **State modification test**
- [ ] Callback that modifies ODE parameters
- [ ] Verify effect persists
- [ ] Integration continues correctly
### Benchmarking
- [ ] Compare overhead vs no callbacks
- [ ] Should be minimal (just boolean check)
- [ ] Compare vs continuous callback for same logical event
- [ ] Discrete should be faster
### Documentation
- [ ] Docstring explaining discrete vs continuous
- [ ] When to use each type
- [ ] Examples:
- [ ] Stop at specific time
- [ ] Parameter update every N time units
- [ ] Terminate when condition met
- [ ] Integration with CallbackSet (future)
## Testing Requirements
### Stop at Time Test
```rust
fn test_stop_at_time() {
let params = ();
fn derivative(_t: f64, y: Vector1<f64>, _p: &()) -> Vector1<f64> {
Vector1::new(y[0])
}
let ode = ODE::new(&derivative, 0.0, 10.0, Vector1::new(1.0), ());
let dp45 = DormandPrince45::new();
let controller = PIController::default();
let stop_callback = DiscreteCallback::single(
&|t: f64, _y, _p| t >= 5.0,
&stop,
);
let mut problem = Problem::new(ode, dp45, controller)
.with_discrete_callback(stop_callback);
let solution = problem.solve();
// Should stop at first step after t=5.0
assert!(solution.times.last().unwrap() >= &5.0);
assert!(solution.times.last().unwrap() < &5.5); // Reasonable step size
}
```
### Parameter Modification Test
```rust
// Callback that changes parameter at t=5.0
// Verify slope of solution changes at that point
```
## References
1. **Julia Implementation**:
- `DiffEqCallbacks.jl/src/discrete_callbacks.jl`
- `OrdinaryDiffEq.jl` - check order of callback evaluation
2. **Design Patterns**:
- "Event Handling in DifferentialEquations.jl"
- DifferentialEquations.jl documentation on callback types
3. **Use Cases**:
- Sundials documentation on user-supplied functions
- MATLAB ODE event handling
## Complexity Estimate
**Effort**: Small (4-6 hours)
- Relatively simple addition
- Similar structure to existing continuous callbacks
- Main work is integration and testing
**Risk**: Low
- Straightforward concept
- Minimal changes to solver core
- Easy to test
## Success Criteria
- [ ] DiscreteCallback struct defined and documented
- [ ] Integrated into Problem solve loop
- [ ] Single-trigger functionality works correctly
- [ ] Can combine with continuous callbacks
- [ ] All tests pass
- [ ] Performance overhead < 5%
- [ ] Documentation with examples
## Future Enhancements
- [ ] CallbackSet for managing multiple callbacks
- [ ] Priority/ordering for callback execution
- [ ] PresetTimeCallback (fires at specific predetermined times)
- [ ] Integration with save points (saveat)
- [ ] Callback composition and chaining

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# Feature: CallbackSet
## Overview
CallbackSet allows composing multiple callbacks (both continuous and discrete) with controlled ordering and execution priority. Essential for complex simulations with multiple events.
## Why This Feature Matters
- Manage multiple callbacks cleanly
- Control execution order
- Enable/disable callbacks dynamically
- Foundation for advanced callback patterns
## Dependencies
- Discrete callbacks (feature #5)
- Continuous callbacks (already implemented)
## Implementation Approach
```rust
pub struct CallbackSet<'a, const D: usize, P> {
continuous_callbacks: Vec<ContinuousCallback<'a, D, P>>,
discrete_callbacks: Vec<DiscreteCallback<'a, D, P>>,
// Optional: priority/ordering information
}
```
## Implementation Tasks
- [ ] Define CallbackSet struct
- [ ] Builder pattern for adding callbacks
- [ ] Execution order management
- [ ] Integration with Problem solve loop
- [ ] Testing with multiple callbacks
- [ ] Documentation
## Complexity Estimate
**Effort**: Small (3-4 hours)
**Risk**: Low

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# Feature: Saveat Functionality
## Overview
Save solution at specific timepoints
## Why This Feature Matters
(To be detailed)
## Dependencies
(To be detailed)
## Implementation Approach
(To be detailed)
## Implementation Tasks
- [ ] Core implementation
- [ ] Integration with existing code
- [ ] Testing
- [ ] Documentation
- [ ] Benchmarking
## Testing Requirements
(To be detailed)
## References
1. Julia implementation: OrdinaryDiffEq.jl
2. (Additional references to be added)
## Complexity Estimate
**Effort**: Medium
**Risk**: (To be assessed)
## Success Criteria
- [ ] Implementation complete
- [ ] Tests pass
- [ ] Documentation written
- [ ] Performance acceptable
## Future Enhancements
(To be identified)

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# Feature: FBDF Method
## Overview
Fixed-leading-coefficient BDF for very stiff problems
## Why This Feature Matters
(To be detailed)
## Dependencies
(To be detailed)
## Implementation Approach
(To be detailed)
## Implementation Tasks
- [ ] Core implementation
- [ ] Integration with existing code
- [ ] Testing
- [ ] Documentation
- [ ] Benchmarking
## Testing Requirements
(To be detailed)
## References
1. Julia implementation: OrdinaryDiffEq.jl
2. (Additional references to be added)
## Complexity Estimate
**Effort**: Large
**Risk**: (To be assessed)
## Success Criteria
- [ ] Implementation complete
- [ ] Tests pass
- [ ] Documentation written
- [ ] Performance acceptable
## Future Enhancements
(To be identified)

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# Feature: Rodas4/Rodas5P Methods
## Overview
Higher-order Rosenbrock methods
## Why This Feature Matters
(To be detailed)
## Dependencies
(To be detailed)
## Implementation Approach
(To be detailed)
## Implementation Tasks
- [ ] Core implementation
- [ ] Integration with existing code
- [ ] Testing
- [ ] Documentation
- [ ] Benchmarking
## Testing Requirements
(To be detailed)
## References
1. Julia implementation: OrdinaryDiffEq.jl
2. (Additional references to be added)
## Complexity Estimate
**Effort**: Medium
**Risk**: (To be assessed)
## Success Criteria
- [ ] Implementation complete
- [ ] Tests pass
- [ ] Documentation written
- [ ] Performance acceptable
## Future Enhancements
(To be identified)

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# Feature: Auto-Switching & Stiffness Detection
## Overview
Automatic algorithm switching detects when a problem transitions between stiff and non-stiff regimes and switches to the appropriate solver automatically. This is one of the most powerful features for robust, user-friendly ODE solving.
**Key Characteristics:**
- Automatic stiffness detection via eigenvalue estimation
- Seamless switching between non-stiff and stiff solvers
- CompositeAlgorithm infrastructure
- Configurable switching criteria
- Basis for DefaultODEAlgorithm (solve without specifying algorithm)
## Why This Feature Matters
- **User-friendly**: User doesn't need to know if problem is stiff
- **Robustness**: Handles problems with changing character
- **Efficiency**: Uses fast explicit methods when possible, switches to implicit when needed
- **Production-ready**: Essential for general-purpose library
- **Real problems**: Many problems are "mildly stiff" or transiently stiff
## Dependencies
### Required
- [ ] At least one stiff solver (Rosenbrock23 or FBDF)
- [ ] At least two non-stiff solvers (have DP5, Tsit5)
- [ ] BS3 recommended for completeness
### Recommended
- [ ] Vern7 for high-accuracy non-stiff
- [ ] Rodas4 or Rodas5P for high-accuracy stiff
- [ ] Multiple controllers (PI, PID)
## Implementation Approach
### Stiffness Detection
**Eigenvalue Estimation**:
```
ρ = ||δf|| / ||δy||
```
Where:
- δy = y_{n+1} - y_n
- δf = f(t_{n+1}, y_{n+1}) - f(t_n, y_n)
- ρ approximates spectral radius of Jacobian
**Stiffness ratio**:
```
S = |ρ * h| / stability_region_size
```
If S > tolerance (e.g., 1.0), problem is stiff.
### Algorithm Switching Logic
1. **Detect stiffness** every few steps
2. **Switch condition**: Stiffness detected for N consecutive steps
3. **Switch back**: Non-stiffness detected for M consecutive steps
4. **Hysteresis**: N < M to avoid chattering
Typical values:
- N = 3-5 (switch to stiff solver)
- M = 25-50 (switch back to non-stiff)
### CompositeAlgorithm Structure
```rust
pub struct CompositeAlgorithm<NonStiff, Stiff> {
pub nonstiff_alg: NonStiff,
pub stiff_alg: Stiff,
pub choice_function: AutoSwitchCache,
}
pub struct AutoSwitchCache {
pub current_algorithm: AlgorithmChoice,
pub consecutive_stiff: usize,
pub consecutive_nonstiff: usize,
pub switch_to_stiff_threshold: usize,
pub switch_to_nonstiff_threshold: usize,
pub stiffness_tolerance: f64,
}
pub enum AlgorithmChoice {
NonStiff,
Stiff,
}
```
### Implementation Challenges
1. **State transfer**: When switching, need to transfer state cleanly
2. **Controller state**: Each algorithm may have different controller state
3. **Interpolation**: Dense output from previous algorithm
4. **First step**: Which algorithm to start with?
## Implementation Tasks
### Core Infrastructure
- [ ] Define `CompositeAlgorithm` struct
- [ ] Generic over two integrator types
- [ ] Store both algorithms
- [ ] Store switching logic state
- [ ] Define `AutoSwitchCache`
- [ ] Current algorithm choice
- [ ] Consecutive step counters
- [ ] Thresholds
- [ ] Stiffness tolerance
- [ ] Implement switching logic
- [ ] Eigenvalue estimation function
- [ ] Stiffness detection
- [ ] Decision to switch
- [ ] Reset counters appropriately
### Integrator Changes
- [ ] Modify `Problem` to work with composite algorithms
- [ ] May need `IntegratorEnum` or dynamic dispatch
- [ ] Or: make Problem generic and handle in solve loop
- [ ] State transfer mechanism
- [ ] Transfer y, t from one integrator to other
- [ ] Transfer/reset controller state
- [ ] Clear interpolation data
- [ ] Solve loop modifications
- [ ] Check for switch every N steps
- [ ] Perform switch if needed
- [ ] Continue with new algorithm
### Eigenvalue Estimation
- [ ] Implement basic estimator
- [ ] Track previous f evaluation
- [ ] Compute ρ = ||δf|| / ||δy||
- [ ] Update estimate smoothly (exponential moving average)
- [ ] Handle edge cases
- [ ] Very small ||δy||
- [ ] First step (no history)
- [ ] After callback event
### Default Algorithm
- [ ] `AutoAlgSwitch` function/constructor
- [ ] Takes tuple of non-stiff algorithms
- [ ] Takes tuple of stiff algorithms
- [ ] Returns CompositeAlgorithm
- [ ] With default switching parameters
- [ ] `DefaultODEAlgorithm` (future)
- [ ] Analyzes problem
- [ ] Selects algorithms based on size, tolerance
- [ ] Returns configured CompositeAlgorithm
### Testing
- [ ] **Transiently stiff problem**
- [ ] Starts non-stiff, becomes stiff, then non-stiff again
- [ ] Example: Van der Pol with time-varying μ
- [ ] Verify switches at right times
- [ ] Verify solution accuracy throughout
- [ ] **Always non-stiff problem**
- [ ] Should never switch to stiff solver
- [ ] Verify minimal overhead
- [ ] **Always stiff problem**
- [ ] Should switch to stiff early
- [ ] Stay on stiff solver
- [ ] **Chattering prevention**
- [ ] Problem near stiffness boundary
- [ ] Verify doesn't switch back and forth rapidly
- [ ] Hysteresis should prevent chattering
- [ ] **State transfer test**
- [ ] Switch mid-integration
- [ ] Verify no discontinuity in solution
- [ ] Interpolation works across switch
- [ ] **Comparison test**
- [ ] Run transient stiff problem three ways:
- [ ] Auto-switching
- [ ] Non-stiff only (should fail or be very slow)
- [ ] Stiff only (should work but possibly slower)
- [ ] Auto-switching should be nearly optimal
### Benchmarking
- [ ] ROBER problem (chemistry, transiently stiff)
- [ ] HIRES problem (atmospheric chemistry)
- [ ] Compare to manual algorithm selection
- [ ] Measure switching overhead
### Documentation
- [ ] Explain stiffness detection
- [ ] Document switching thresholds
- [ ] When auto-switching helps vs hurts
- [ ] Examples with different problem types
- [ ] How to configure switching parameters
## Testing Requirements
### Transient Stiffness Test
Van der Pol oscillator with time-varying stiffness:
```rust
// μ(t) = 100 for t < 20
// μ(t) = 1 for 20 <= t < 40
// μ(t) = 100 for t >= 40
```
Expected behavior:
- Start with non-stiff (or quickly switch to stiff)
- Switch to non-stiff around t=20
- Switch back to stiff around t=40
- Solution remains accurate throughout
Track:
- When switches occur
- Number of switches
- Total steps with each algorithm
### ROBER Problem
Robertson chemical kinetics:
```
y1' = -0.04*y1 + 1e4*y2*y3
y2' = 0.04*y1 - 1e4*y2*y3 - 3e7*y2²
y3' = 3e7*y2²
```
Very stiff initially, becomes less stiff.
Expected: Should start with (or quickly switch to) stiff solver.
## References
1. **Stiffness Detection**:
- Shampine, L.F. (1977)
- "Stiffness and Non-stiff Differential Equation Solvers, II"
- Applied Numerical Mathematics
2. **Auto-switching Algorithms**:
- Hairer & Wanner (1996), "Solving ODEs II", Section IV.3
- Discussion of when to use stiff solvers
3. **Julia Implementation**:
- `OrdinaryDiffEq.jl/lib/OrdinaryDiffEqDefault/src/default_alg.jl`
- `AutoAlgSwitch` and `default_autoswitch` functions
4. **MATLAB's ode45/ode15s switching**:
- MATLAB documentation on automatic solver selection
## Complexity Estimate
**Effort**: Large (15-25 hours)
- Composite algorithm infrastructure: 6-8 hours
- Stiffness detection: 4-6 hours
- Switching logic and state transfer: 5-8 hours
- Testing and tuning: 4-6 hours
**Risk**: Medium-High
- Complexity in state transfer
- Getting switching criteria right requires tuning
- Interaction with controllers needs care
- Edge cases (callbacks during switch, etc.)
## Success Criteria
- [ ] Handles transiently stiff problems automatically
- [ ] Switches at appropriate times
- [ ] No chattering between algorithms
- [ ] Solution accuracy maintained across switches
- [ ] Overhead < 10% on problems that don't need switching
- [ ] Performance within 20% of manual optimal selection
- [ ] Documentation complete with examples
- [ ] Robust to edge cases
## Future Enhancements
- [ ] More sophisticated stiffness detection
- [ ] Multiple detection methods
- [ ] Learning from past behavior
- [ ] Multi-algorithm selection
- [ ] More than 2 algorithms (low/medium/high accuracy)
- [ ] Tolerance-based selection
- [ ] Automatic tolerance selection
- [ ] Problem analysis at start
- [ ] Estimate problem size effect
- [ ] Sparsity detection
- [ ] Initial algorithm recommendation
- [ ] DefaultODEAlgorithm with full analysis
- [ ] Based on problem size, tolerance, mass matrix, etc.

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# Feature: Default Algorithm Selection
## Overview
Smart defaults based on problem characteristics
## Why This Feature Matters
(To be detailed)
## Dependencies
(To be detailed)
## Implementation Approach
(To be detailed)
## Implementation Tasks
- [ ] Core implementation
- [ ] Integration with existing code
- [ ] Testing
- [ ] Documentation
- [ ] Benchmarking
## Testing Requirements
(To be detailed)
## References
1. Julia implementation: OrdinaryDiffEq.jl
2. (Additional references to be added)
## Complexity Estimate
**Effort**: Medium
**Risk**: (To be assessed)
## Success Criteria
- [ ] Implementation complete
- [ ] Tests pass
- [ ] Documentation written
- [ ] Performance acceptable
## Future Enhancements
(To be identified)

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# Feature: Automatic Initial Step Size
## Overview
Algorithm to determine good initial dt
## Why This Feature Matters
(To be detailed)
## Dependencies
(To be detailed)
## Implementation Approach
(To be detailed)
## Implementation Tasks
- [ ] Core implementation
- [ ] Integration with existing code
- [ ] Testing
- [ ] Documentation
- [ ] Benchmarking
## Testing Requirements
(To be detailed)
## References
1. Julia implementation: OrdinaryDiffEq.jl
2. (Additional references to be added)
## Complexity Estimate
**Effort**: Medium
**Risk**: (To be assessed)
## Success Criteria
- [ ] Implementation complete
- [ ] Tests pass
- [ ] Documentation written
- [ ] Performance acceptable
## Future Enhancements
(To be identified)

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# Feature: PresetTimeCallback
## Overview
Callbacks at predetermined times
## Why This Feature Matters
(To be detailed)
## Dependencies
(To be detailed)
## Implementation Approach
(To be detailed)
## Implementation Tasks
- [ ] Core implementation
- [ ] Integration with existing code
- [ ] Testing
- [ ] Documentation
- [ ] Benchmarking
## Testing Requirements
(To be detailed)
## References
1. Julia implementation: OrdinaryDiffEq.jl
2. (Additional references to be added)
## Complexity Estimate
**Effort**: Small
**Risk**: (To be assessed)
## Success Criteria
- [ ] Implementation complete
- [ ] Tests pass
- [ ] Documentation written
- [ ] Performance acceptable
## Future Enhancements
(To be identified)

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# Feature: TerminateSteadyState
## Overview
Auto-detect steady state
## Why This Feature Matters
(To be detailed)
## Dependencies
(To be detailed)
## Implementation Approach
(To be detailed)
## Implementation Tasks
- [ ] Core implementation
- [ ] Integration with existing code
- [ ] Testing
- [ ] Documentation
- [ ] Benchmarking
## Testing Requirements
(To be detailed)
## References
1. Julia implementation: OrdinaryDiffEq.jl
2. (Additional references to be added)
## Complexity Estimate
**Effort**: Small
**Risk**: (To be assessed)
## Success Criteria
- [ ] Implementation complete
- [ ] Tests pass
- [ ] Documentation written
- [ ] Performance acceptable
## Future Enhancements
(To be identified)

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# Feature: SavingCallback
## Overview
Custom saving logic
## Why This Feature Matters
(To be detailed)
## Dependencies
(To be detailed)
## Implementation Approach
(To be detailed)
## Implementation Tasks
- [ ] Core implementation
- [ ] Integration with existing code
- [ ] Testing
- [ ] Documentation
- [ ] Benchmarking
## Testing Requirements
(To be detailed)
## References
1. Julia implementation: OrdinaryDiffEq.jl
2. (Additional references to be added)
## Complexity Estimate
**Effort**: Small
**Risk**: (To be assessed)
## Success Criteria
- [ ] Implementation complete
- [ ] Tests pass
- [ ] Documentation written
- [ ] Performance acceptable
## Future Enhancements
(To be identified)

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# Feature: Linear Solver Infrastructure
## Overview
Generic linear solver interface and dense LU
## Why This Feature Matters
(To be detailed)
## Dependencies
(To be detailed)
## Implementation Approach
(To be detailed)
## Implementation Tasks
- [ ] Core implementation
- [ ] Integration with existing code
- [ ] Testing
- [ ] Documentation
- [ ] Benchmarking
## Testing Requirements
(To be detailed)
## References
1. Julia implementation: OrdinaryDiffEq.jl
2. (Additional references to be added)
## Complexity Estimate
**Effort**: Large
**Risk**: (To be assessed)
## Success Criteria
- [ ] Implementation complete
- [ ] Tests pass
- [ ] Documentation written
- [ ] Performance acceptable
## Future Enhancements
(To be identified)

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# Feature: Jacobian Computation
## Overview
Finite difference and auto-diff Jacobians
## Why This Feature Matters
(To be detailed)
## Dependencies
(To be detailed)
## Implementation Approach
(To be detailed)
## Implementation Tasks
- [ ] Core implementation
- [ ] Integration with existing code
- [ ] Testing
- [ ] Documentation
- [ ] Benchmarking
## Testing Requirements
(To be detailed)
## References
1. Julia implementation: OrdinaryDiffEq.jl
2. (Additional references to be added)
## Complexity Estimate
**Effort**: Large
**Risk**: (To be assessed)
## Success Criteria
- [ ] Implementation complete
- [ ] Tests pass
- [ ] Documentation written
- [ ] Performance acceptable
## Future Enhancements
(To be identified)

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# Feature: Low-Storage Runge-Kutta
## Overview
2N/3N storage variants for large systems
## Why This Feature Matters
(To be detailed)
## Dependencies
(To be detailed)
## Implementation Approach
(To be detailed)
## Implementation Tasks
- [ ] Core implementation
- [ ] Integration with existing code
- [ ] Testing
- [ ] Documentation
- [ ] Benchmarking
## Testing Requirements
(To be detailed)
## References
1. Julia implementation: OrdinaryDiffEq.jl
2. (Additional references to be added)
## Complexity Estimate
**Effort**: Medium
**Risk**: (To be assessed)
## Success Criteria
- [ ] Implementation complete
- [ ] Tests pass
- [ ] Documentation written
- [ ] Performance acceptable
## Future Enhancements
(To be identified)

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# Feature: SSP Methods
## Overview
Strong Stability Preserving methods
## Why This Feature Matters
(To be detailed)
## Dependencies
(To be detailed)
## Implementation Approach
(To be detailed)
## Implementation Tasks
- [ ] Core implementation
- [ ] Integration with existing code
- [ ] Testing
- [ ] Documentation
- [ ] Benchmarking
## Testing Requirements
(To be detailed)
## References
1. Julia implementation: OrdinaryDiffEq.jl
2. (Additional references to be added)
## Complexity Estimate
**Effort**: Medium
**Risk**: (To be assessed)
## Success Criteria
- [ ] Implementation complete
- [ ] Tests pass
- [ ] Documentation written
- [ ] Performance acceptable
## Future Enhancements
(To be identified)

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# Feature: Symplectic Integrators
## Overview
Verlet, Leapfrog, KahanLi for Hamiltonian systems
## Why This Feature Matters
(To be detailed)
## Dependencies
(To be detailed)
## Implementation Approach
(To be detailed)
## Implementation Tasks
- [ ] Core implementation
- [ ] Integration with existing code
- [ ] Testing
- [ ] Documentation
- [ ] Benchmarking
## Testing Requirements
(To be detailed)
## References
1. Julia implementation: OrdinaryDiffEq.jl
2. (Additional references to be added)
## Complexity Estimate
**Effort**: Medium
**Risk**: (To be assessed)
## Success Criteria
- [ ] Implementation complete
- [ ] Tests pass
- [ ] Documentation written
- [ ] Performance acceptable
## Future Enhancements
(To be identified)

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# Feature: Verner Methods Suite
## Overview
Vern6, Vern8, Vern9
## Why This Feature Matters
(To be detailed)
## Dependencies
(To be detailed)
## Implementation Approach
(To be detailed)
## Implementation Tasks
- [ ] Core implementation
- [ ] Integration with existing code
- [ ] Testing
- [ ] Documentation
- [ ] Benchmarking
## Testing Requirements
(To be detailed)
## References
1. Julia implementation: OrdinaryDiffEq.jl
2. (Additional references to be added)
## Complexity Estimate
**Effort**: Medium
**Risk**: (To be assessed)
## Success Criteria
- [ ] Implementation complete
- [ ] Tests pass
- [ ] Documentation written
- [ ] Performance acceptable
## Future Enhancements
(To be identified)

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# Feature: SDIRK Methods
## Overview
KenCarp3/4/5 for stiff problems
## Why This Feature Matters
(To be detailed)
## Dependencies
(To be detailed)
## Implementation Approach
(To be detailed)
## Implementation Tasks
- [ ] Core implementation
- [ ] Integration with existing code
- [ ] Testing
- [ ] Documentation
- [ ] Benchmarking
## Testing Requirements
(To be detailed)
## References
1. Julia implementation: OrdinaryDiffEq.jl
2. (Additional references to be added)
## Complexity Estimate
**Effort**: Large
**Risk**: (To be assessed)
## Success Criteria
- [ ] Implementation complete
- [ ] Tests pass
- [ ] Documentation written
- [ ] Performance acceptable
## Future Enhancements
(To be identified)

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# Feature: Exponential Integrators
## Overview
Exp4, EPIRK4 for semi-linear problems
## Why This Feature Matters
(To be detailed)
## Dependencies
(To be detailed)
## Implementation Approach
(To be detailed)
## Implementation Tasks
- [ ] Core implementation
- [ ] Integration with existing code
- [ ] Testing
- [ ] Documentation
- [ ] Benchmarking
## Testing Requirements
(To be detailed)
## References
1. Julia implementation: OrdinaryDiffEq.jl
2. (Additional references to be added)
## Complexity Estimate
**Effort**: Large
**Risk**: (To be assessed)
## Success Criteria
- [ ] Implementation complete
- [ ] Tests pass
- [ ] Documentation written
- [ ] Performance acceptable
## Future Enhancements
(To be identified)

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# Feature: Extrapolation Methods
## Overview
Richardson extrapolation with adaptive order
## Why This Feature Matters
(To be detailed)
## Dependencies
(To be detailed)
## Implementation Approach
(To be detailed)
## Implementation Tasks
- [ ] Core implementation
- [ ] Integration with existing code
- [ ] Testing
- [ ] Documentation
- [ ] Benchmarking
## Testing Requirements
(To be detailed)
## References
1. Julia implementation: OrdinaryDiffEq.jl
2. (Additional references to be added)
## Complexity Estimate
**Effort**: Large
**Risk**: (To be assessed)
## Success Criteria
- [ ] Implementation complete
- [ ] Tests pass
- [ ] Documentation written
- [ ] Performance acceptable
## Future Enhancements
(To be identified)

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# Feature: Stabilized Methods
## Overview
ROCK2, ROCK4, RKC for mildly stiff
## Why This Feature Matters
(To be detailed)
## Dependencies
(To be detailed)
## Implementation Approach
(To be detailed)
## Implementation Tasks
- [ ] Core implementation
- [ ] Integration with existing code
- [ ] Testing
- [ ] Documentation
- [ ] Benchmarking
## Testing Requirements
(To be detailed)
## References
1. Julia implementation: OrdinaryDiffEq.jl
2. (Additional references to be added)
## Complexity Estimate
**Effort**: Medium
**Risk**: (To be assessed)
## Success Criteria
- [ ] Implementation complete
- [ ] Tests pass
- [ ] Documentation written
- [ ] Performance acceptable
## Future Enhancements
(To be identified)

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# Feature: I Controller
## Overview
Basic integral controller
## Why This Feature Matters
(To be detailed)
## Dependencies
(To be detailed)
## Implementation Approach
(To be detailed)
## Implementation Tasks
- [ ] Core implementation
- [ ] Integration with existing code
- [ ] Testing
- [ ] Documentation
- [ ] Benchmarking
## Testing Requirements
(To be detailed)
## References
1. Julia implementation: OrdinaryDiffEq.jl
2. (Additional references to be added)
## Complexity Estimate
**Effort**: Small
**Risk**: (To be assessed)
## Success Criteria
- [ ] Implementation complete
- [ ] Tests pass
- [ ] Documentation written
- [ ] Performance acceptable
## Future Enhancements
(To be identified)

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# Feature: Predictive Controller
## Overview
Advanced predictive controller
## Why This Feature Matters
(To be detailed)
## Dependencies
(To be detailed)
## Implementation Approach
(To be detailed)
## Implementation Tasks
- [ ] Core implementation
- [ ] Integration with existing code
- [ ] Testing
- [ ] Documentation
- [ ] Benchmarking
## Testing Requirements
(To be detailed)
## References
1. Julia implementation: OrdinaryDiffEq.jl
2. (Additional references to be added)
## Complexity Estimate
**Effort**: Medium
**Risk**: (To be assessed)
## Success Criteria
- [ ] Implementation complete
- [ ] Tests pass
- [ ] Documentation written
- [ ] Performance acceptable
## Future Enhancements
(To be identified)

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# Feature: VectorContinuousCallback
## Overview
Multiple simultaneous events
## Why This Feature Matters
(To be detailed)
## Dependencies
(To be detailed)
## Implementation Approach
(To be detailed)
## Implementation Tasks
- [ ] Core implementation
- [ ] Integration with existing code
- [ ] Testing
- [ ] Documentation
- [ ] Benchmarking
## Testing Requirements
(To be detailed)
## References
1. Julia implementation: OrdinaryDiffEq.jl
2. (Additional references to be added)
## Complexity Estimate
**Effort**: Medium
**Risk**: (To be assessed)
## Success Criteria
- [ ] Implementation complete
- [ ] Tests pass
- [ ] Documentation written
- [ ] Performance acceptable
## Future Enhancements
(To be identified)

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# Feature: PositiveDomain
## Overview
Enforce positivity constraints
## Why This Feature Matters
(To be detailed)
## Dependencies
(To be detailed)
## Implementation Approach
(To be detailed)
## Implementation Tasks
- [ ] Core implementation
- [ ] Integration with existing code
- [ ] Testing
- [ ] Documentation
- [ ] Benchmarking
## Testing Requirements
(To be detailed)
## References
1. Julia implementation: OrdinaryDiffEq.jl
2. (Additional references to be added)
## Complexity Estimate
**Effort**: Small
**Risk**: (To be assessed)
## Success Criteria
- [ ] Implementation complete
- [ ] Tests pass
- [ ] Documentation written
- [ ] Performance acceptable
## Future Enhancements
(To be identified)

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# Feature: ManifoldProjection
## Overview
Project onto constraint manifolds
## Why This Feature Matters
(To be detailed)
## Dependencies
(To be detailed)
## Implementation Approach
(To be detailed)
## Implementation Tasks
- [ ] Core implementation
- [ ] Integration with existing code
- [ ] Testing
- [ ] Documentation
- [ ] Benchmarking
## Testing Requirements
(To be detailed)
## References
1. Julia implementation: OrdinaryDiffEq.jl
2. (Additional references to be added)
## Complexity Estimate
**Effort**: Medium
**Risk**: (To be assessed)
## Success Criteria
- [ ] Implementation complete
- [ ] Tests pass
- [ ] Documentation written
- [ ] Performance acceptable
## Future Enhancements
(To be identified)

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# Feature: Nonlinear Solver Infrastructure
## Overview
Newton and quasi-Newton methods
## Why This Feature Matters
(To be detailed)
## Dependencies
(To be detailed)
## Implementation Approach
(To be detailed)
## Implementation Tasks
- [ ] Core implementation
- [ ] Integration with existing code
- [ ] Testing
- [ ] Documentation
- [ ] Benchmarking
## Testing Requirements
(To be detailed)
## References
1. Julia implementation: OrdinaryDiffEq.jl
2. (Additional references to be added)
## Complexity Estimate
**Effort**: Large
**Risk**: (To be assessed)
## Success Criteria
- [ ] Implementation complete
- [ ] Tests pass
- [ ] Documentation written
- [ ] Performance acceptable
## Future Enhancements
(To be identified)

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# Feature: Krylov Linear Solvers
## Overview
GMRES, BiCGStab for large sparse systems
## Why This Feature Matters
(To be detailed)
## Dependencies
(To be detailed)
## Implementation Approach
(To be detailed)
## Implementation Tasks
- [ ] Core implementation
- [ ] Integration with existing code
- [ ] Testing
- [ ] Documentation
- [ ] Benchmarking
## Testing Requirements
(To be detailed)
## References
1. Julia implementation: OrdinaryDiffEq.jl
2. (Additional references to be added)
## Complexity Estimate
**Effort**: Large
**Risk**: (To be assessed)
## Success Criteria
- [ ] Implementation complete
- [ ] Tests pass
- [ ] Documentation written
- [ ] Performance acceptable
## Future Enhancements
(To be identified)

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# Feature: Preconditioners
## Overview
ILU, Jacobi preconditioners
## Why This Feature Matters
(To be detailed)
## Dependencies
(To be detailed)
## Implementation Approach
(To be detailed)
## Implementation Tasks
- [ ] Core implementation
- [ ] Integration with existing code
- [ ] Testing
- [ ] Documentation
- [ ] Benchmarking
## Testing Requirements
(To be detailed)
## References
1. Julia implementation: OrdinaryDiffEq.jl
2. (Additional references to be added)
## Complexity Estimate
**Effort**: Large
**Risk**: (To be assessed)
## Success Criteria
- [ ] Implementation complete
- [ ] Tests pass
- [ ] Documentation written
- [ ] Performance acceptable
## Future Enhancements
(To be identified)

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# Feature: FSAL Optimization
## Overview
First-Same-As-Last function reuse
## Why This Feature Matters
(To be detailed)
## Dependencies
(To be detailed)
## Implementation Approach
(To be detailed)
## Implementation Tasks
- [ ] Core implementation
- [ ] Integration with existing code
- [ ] Testing
- [ ] Documentation
- [ ] Benchmarking
## Testing Requirements
(To be detailed)
## References
1. Julia implementation: OrdinaryDiffEq.jl
2. (Additional references to be added)
## Complexity Estimate
**Effort**: Small
**Risk**: (To be assessed)
## Success Criteria
- [ ] Implementation complete
- [ ] Tests pass
- [ ] Documentation written
- [ ] Performance acceptable
## Future Enhancements
(To be identified)

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# Feature: Custom Norms
## Overview
User-definable error norms
## Why This Feature Matters
(To be detailed)
## Dependencies
(To be detailed)
## Implementation Approach
(To be detailed)
## Implementation Tasks
- [ ] Core implementation
- [ ] Integration with existing code
- [ ] Testing
- [ ] Documentation
- [ ] Benchmarking
## Testing Requirements
(To be detailed)
## References
1. Julia implementation: OrdinaryDiffEq.jl
2. (Additional references to be added)
## Complexity Estimate
**Effort**: Small
**Risk**: (To be assessed)
## Success Criteria
- [ ] Implementation complete
- [ ] Tests pass
- [ ] Documentation written
- [ ] Performance acceptable
## Future Enhancements
(To be identified)

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# Feature: Step/Stage Limiting
## Overview
Limit state values during integration
## Why This Feature Matters
(To be detailed)
## Dependencies
(To be detailed)
## Implementation Approach
(To be detailed)
## Implementation Tasks
- [ ] Core implementation
- [ ] Integration with existing code
- [ ] Testing
- [ ] Documentation
- [ ] Benchmarking
## Testing Requirements
(To be detailed)
## References
1. Julia implementation: OrdinaryDiffEq.jl
2. (Additional references to be added)
## Complexity Estimate
**Effort**: Small
**Risk**: (To be assessed)
## Success Criteria
- [ ] Implementation complete
- [ ] Tests pass
- [ ] Documentation written
- [ ] Performance acceptable
## Future Enhancements
(To be identified)

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use nalgebra::SVector;
use super::super::ode::ODE;
use super::Integrator;
/// Bogacki-Shampine 3/2 integrator trait for tableau coefficients
pub trait BS3Integrator<'a> {
const A: &'a [f64];
const B: &'a [f64];
const B_ERROR: &'a [f64];
const C: &'a [f64];
}
/// Bogacki-Shampine 3(2) method
///
/// A 3rd order explicit Runge-Kutta method with an embedded 2nd order method for
/// error estimation. This method is efficient for moderate accuracy requirements
/// (tolerances around 1e-3 to 1e-6) and uses fewer stages than Dormand-Prince 4(5).
///
/// # Characteristics
/// - Order: 3(2) - 3rd order solution with 2nd order error estimate
/// - Stages: 4
/// - FSAL: Yes (First Same As Last - reuses last function evaluation)
/// - Adaptive: Yes
/// - Dense output: 3rd order Hermite interpolation
///
/// # When to use BS3
/// - Problems requiring moderate accuracy (rtol ~ 1e-3 to 1e-6)
/// - When function evaluations are expensive (fewer stages than DP5)
/// - Non-stiff problems
///
/// # Example
/// ```rust
/// use ordinary_diffeq::prelude::*;
/// use nalgebra::Vector1;
///
/// let params = ();
/// fn derivative(_t: f64, y: Vector1<f64>, _p: &()) -> Vector1<f64> {
/// Vector1::new(-y[0])
/// }
///
/// let y0 = Vector1::new(1.0);
/// let ode = ODE::new(&derivative, 0.0, 5.0, y0, ());
/// let bs3 = BS3::new().a_tol(1e-6).r_tol(1e-4);
/// let controller = PIController::default();
///
/// let mut problem = Problem::new(ode, bs3, controller);
/// let solution = problem.solve();
/// ```
///
/// # References
/// - Bogacki, P. and Shampine, L.F. (1989), "A 3(2) pair of Runge-Kutta formulas",
/// Applied Mathematics Letters, Vol. 2, No. 4, pp. 321-325
#[derive(Debug, Clone, Copy)]
pub struct BS3<const D: usize> {
a_tol: SVector<f64, D>,
r_tol: f64,
}
impl<const D: usize> BS3<D>
where
BS3<D>: Integrator<D>,
{
/// Create a new BS3 integrator with default tolerances
///
/// Default: atol = 1e-8, rtol = 1e-8
pub fn new() -> Self {
Self {
a_tol: SVector::<f64, D>::from_element(1e-8),
r_tol: 1e-8,
}
}
/// Set absolute tolerance (same value for all components)
pub fn a_tol(mut self, a_tol: f64) -> Self {
self.a_tol = SVector::<f64, D>::from_element(a_tol);
self
}
/// Set absolute tolerance (different value per component)
pub fn a_tol_full(mut self, a_tol: SVector<f64, D>) -> Self {
self.a_tol = a_tol;
self
}
/// Set relative tolerance
pub fn r_tol(mut self, r_tol: f64) -> Self {
self.r_tol = r_tol;
self
}
}
impl<'a, const D: usize> BS3Integrator<'a> for BS3<D> {
// Butcher tableau for BS3
// The A matrix is stored in lower-triangular form as a flat array
// Row 1: []
// Row 2: [1/2]
// Row 3: [0, 3/4]
// Row 4: [2/9, 1/3, 4/9]
const A: &'a [f64] = &[
1.0 / 2.0, // a[1,0]
0.0, // a[2,0]
3.0 / 4.0, // a[2,1]
2.0 / 9.0, // a[3,0]
1.0 / 3.0, // a[3,1]
4.0 / 9.0, // a[3,2]
];
// Solution weights (3rd order)
const B: &'a [f64] = &[
2.0 / 9.0, // b[0]
1.0 / 3.0, // b[1]
4.0 / 9.0, // b[2]
0.0, // b[3] - FSAL property: this is zero
];
// Error estimate weights (difference between 3rd and 2nd order)
const B_ERROR: &'a [f64] = &[
2.0 / 9.0 - 7.0 / 24.0, // b[0] - b*[0]
1.0 / 3.0 - 1.0 / 4.0, // b[1] - b*[1]
4.0 / 9.0 - 1.0 / 3.0, // b[2] - b*[2]
0.0 - 1.0 / 8.0, // b[3] - b*[3]
];
// Stage times
const C: &'a [f64] = &[
0.0, // c[0]
1.0 / 2.0, // c[1]
3.0 / 4.0, // c[2]
1.0, // c[3]
];
}
impl<'a, const D: usize> Integrator<D> for BS3<D>
where
BS3<D>: BS3Integrator<'a>,
{
const ORDER: usize = 3;
const STAGES: usize = 4;
const ADAPTIVE: bool = true;
const DENSE: bool = true;
fn step<P>(
&self,
ode: &ODE<D, P>,
h: f64,
) -> (SVector<f64, D>, Option<f64>, Option<Vec<SVector<f64, D>>>) {
// Allocate storage for the 4 stages
let mut k: Vec<SVector<f64, D>> = vec![SVector::<f64, D>::zeros(); Self::STAGES];
// Stage 1: k1 = f(t, y)
k[0] = (ode.f)(ode.t, ode.y, &ode.params);
// Stage 2: k2 = f(t + c[1]*h, y + h*a[1,0]*k1)
let y2 = ode.y + h * Self::A[0] * k[0];
k[1] = (ode.f)(ode.t + Self::C[1] * h, y2, &ode.params);
// Stage 3: k3 = f(t + c[2]*h, y + h*(a[2,0]*k1 + a[2,1]*k2))
let y3 = ode.y + h * (Self::A[1] * k[0] + Self::A[2] * k[1]);
k[2] = (ode.f)(ode.t + Self::C[2] * h, y3, &ode.params);
// Stage 4: k4 = f(t + c[3]*h, y + h*(a[3,0]*k1 + a[3,1]*k2 + a[3,2]*k3))
let y4 = ode.y + h * (Self::A[3] * k[0] + Self::A[4] * k[1] + Self::A[5] * k[2]);
k[3] = (ode.f)(ode.t + Self::C[3] * h, y4, &ode.params);
// Compute 3rd order solution
let next_y = ode.y + h * (Self::B[0] * k[0] + Self::B[1] * k[1] + Self::B[2] * k[2] + Self::B[3] * k[3]);
// Compute error estimate (difference between 3rd and 2nd order solutions)
let err = h * (Self::B_ERROR[0] * k[0] + Self::B_ERROR[1] * k[1] + Self::B_ERROR[2] * k[2] + Self::B_ERROR[3] * k[3]);
// Compute error norm scaled by tolerance
let tol = self.a_tol + ode.y.abs() * self.r_tol;
let error_norm = (err.component_div(&tol)).norm();
// Store coefficients for dense output (cubic Hermite interpolation)
// BS3 uses standard cubic Hermite interpolation with derivatives at endpoints
// Store: y0, y1, f0=k[0], f1=k[3] (FSAL)
let dense_coeffs = vec![
ode.y, // y0 at start of step
next_y, // y1 at end of step
k[0], // f(t0, y0) - derivative at start
k[3], // f(t1, y1) - derivative at end (FSAL)
];
(next_y, Some(error_norm), Some(dense_coeffs))
}
fn interpolate(
&self,
t_start: f64,
t_end: f64,
dense: &[SVector<f64, D>],
t: f64,
) -> SVector<f64, D> {
// Compute interpolation parameter θ ∈ [0, 1]
let theta = (t - t_start) / (t_end - t_start);
let h = t_end - t_start;
// Cubic Hermite interpolation using values and derivatives at endpoints
// dense[0] = y0 (value at start)
// dense[1] = y1 (value at end)
// dense[2] = f0 (derivative at start)
// dense[3] = f1 (derivative at end)
//
// Standard cubic Hermite formula:
// y(θ) = (1 + 2θ)(1-θ)²*y0 + θ²(3-2θ)*y1 + θ(1-θ)²*h*f0 + θ²(θ-1)*h*f1
//
// Equivalently (Horner form):
// y(θ) = y0 + θ*[h*f0 + θ*(-3*y0 - 2*h*f0 + 3*y1 - h*f1 + θ*(2*y0 + h*f0 - 2*y1 + h*f1))]
let y0 = &dense[0];
let y1 = &dense[1];
let f0 = &dense[2];
let f1 = &dense[3];
let theta2 = theta * theta;
let one_minus_theta = 1.0 - theta;
let one_minus_theta2 = one_minus_theta * one_minus_theta;
// Apply cubic Hermite interpolation formula
(1.0 + 2.0 * theta) * one_minus_theta2 * y0
+ theta2 * (3.0 - 2.0 * theta) * y1
+ theta * one_minus_theta2 * h * f0
+ theta2 * (theta - 1.0) * h * f1
}
}
#[cfg(test)]
mod tests {
use super::*;
use approx::assert_relative_eq;
use nalgebra::Vector1;
#[test]
fn test_bs3_creation() {
let _bs3: BS3<1> = BS3::new();
assert_eq!(BS3::<1>::ORDER, 3);
assert_eq!(BS3::<1>::STAGES, 4);
assert!(BS3::<1>::ADAPTIVE);
assert!(BS3::<1>::DENSE);
}
#[test]
fn test_bs3_step() {
type Params = ();
fn derivative(_t: f64, y: Vector1<f64>, _p: &Params) -> Vector1<f64> {
Vector1::new(y[0]) // y' = y, solution is e^t
}
let y0 = Vector1::new(1.0);
let ode = ODE::new(&derivative, 0.0, 1.0, y0, ());
let bs3 = BS3::new();
let h = 0.001; // Smaller step size for tighter tolerances
let (y_next, err, dense) = bs3.step(&ode, h);
// At t=0.001, exact solution is e^0.001 ≈ 1.0010005001667084
let exact = (0.001_f64).exp();
assert_relative_eq!(y_next[0], exact, max_relative = 1e-6);
// Error should be reasonable for h=0.001
assert!(err.is_some());
// The error estimate is scaled by tolerance, so err < 1 means step is acceptable
assert!(err.unwrap() < 1.0);
// Dense output should be provided
assert!(dense.is_some());
assert_eq!(dense.unwrap().len(), 4);
}
#[test]
fn test_bs3_interpolation() {
type Params = ();
fn derivative(_t: f64, y: Vector1<f64>, _p: &Params) -> Vector1<f64> {
Vector1::new(y[0])
}
let y0 = Vector1::new(1.0);
let ode = ODE::new(&derivative, 0.0, 1.0, y0, ());
let bs3 = BS3::new();
let h = 0.001; // Smaller step size
let (_y_next, _err, dense) = bs3.step(&ode, h);
let dense = dense.unwrap();
// Interpolate at midpoint
let t_mid = 0.0005;
let y_mid = bs3.interpolate(0.0, 0.001, &dense, t_mid);
// Should be close to e^0.0005
let exact = (0.0005_f64).exp();
// Cubic Hermite interpolation should be quite accurate
assert_relative_eq!(y_mid[0], exact, max_relative = 1e-10);
}
#[test]
fn test_bs3_accuracy() {
// Test BS3 on a simple problem with known solution
// y' = -y, y(0) = 1, solution is y(t) = e^(-t)
type Params = ();
fn derivative(_t: f64, y: Vector1<f64>, _p: &Params) -> Vector1<f64> {
Vector1::new(-y[0])
}
let y0 = Vector1::new(1.0);
let bs3 = BS3::new().a_tol(1e-10).r_tol(1e-10);
let h = 0.01;
// Take 100 steps to reach t = 1.0
let mut ode = ODE::new(&derivative, 0.0, 1.0, y0, ());
for _ in 0..100 {
let (y_new, _, _) = bs3.step(&ode, h);
ode.y = y_new;
ode.t += h;
}
// At t=1.0, exact solution is e^(-1) ≈ 0.36787944117
let exact = (-1.0_f64).exp();
assert_relative_eq!(ode.y[0], exact, max_relative = 1e-7);
}
#[test]
fn test_bs3_convergence() {
// Test that BS3 achieves 3rd order convergence
// For a 3rd order method, halving h should reduce error by factor of ~2^3 = 8
type Params = ();
fn derivative(_t: f64, y: Vector1<f64>, _p: &Params) -> Vector1<f64> {
Vector1::new(y[0]) // y' = y, solution is e^t
}
let bs3 = BS3::new();
let t_start = 0.0;
let t_end = 1.0;
let y0 = Vector1::new(1.0);
// Test with different step sizes
let step_sizes = [0.1, 0.05, 0.025];
let mut errors = Vec::new();
for &h in &step_sizes {
let mut ode = ODE::new(&derivative, t_start, t_end, y0, ());
// Take steps until we reach t_end
while ode.t < t_end - 1e-10 {
let (y_new, _, _) = bs3.step(&ode, h);
ode.y = y_new;
ode.t += h;
}
// Compute error at final time
let exact = t_end.exp();
let error = (ode.y[0] - exact).abs();
errors.push(error);
}
// Check convergence rate between consecutive step sizes
for i in 0..errors.len() - 1 {
let ratio = errors[i] / errors[i + 1];
// For order 3, we expect ratio ≈ 2^3 = 8 (since we halve the step size)
// Allow some tolerance due to floating point arithmetic
assert!(
ratio > 6.0 && ratio < 10.0,
"Expected convergence ratio ~8, got {:.2}",
ratio
);
}
// The error should decrease as step size decreases
for i in 0..errors.len() - 1 {
assert!(errors[i] > errors[i + 1]);
}
}
#[test]
fn test_bs3_fsal_property() {
// Test that BS3 correctly implements the FSAL (First Same As Last) property
// The last function evaluation of one step should equal the first of the next
type Params = ();
fn derivative(_t: f64, y: Vector1<f64>, _p: &Params) -> Vector1<f64> {
Vector1::new(2.0 * y[0]) // y' = 2y
}
let y0 = Vector1::new(1.0);
let bs3 = BS3::new();
let h = 0.1;
// First step
let ode1 = ODE::new(&derivative, 0.0, 1.0, y0, ());
let (y_new1, _, dense1) = bs3.step(&ode1, h);
let dense1 = dense1.unwrap();
// Extract f1 from first step (derivative at end of step)
let f1_end = &dense1[3]; // f(t1, y1)
// Second step starts where first ended
let ode2 = ODE::new(&derivative, h, 1.0, y_new1, ());
let (_, _, dense2) = bs3.step(&ode2, h);
let dense2 = dense2.unwrap();
// Extract f0 from second step (derivative at start of step)
let f0_start = &dense2[2]; // f(t0, y0) of second step
// These should be equal (FSAL property)
assert_relative_eq!(f1_end[0], f0_start[0], max_relative = 1e-14);
}
}

1
src/integrator/mod.rs
View File

@@ -2,6 +2,7 @@ use nalgebra::SVector;
use super::ode::ODE; use super::ode::ODE;
pub mod bs3;
pub mod dormand_prince; pub mod dormand_prince;
// pub mod rosenbrock; // pub mod rosenbrock;

1
src/lib.rs
View File

@@ -9,6 +9,7 @@ pub mod problem;
pub mod prelude { pub mod prelude {
pub use super::callback::{stop, Callback}; pub use super::callback::{stop, Callback};
pub use super::controller::PIController; pub use super::controller::PIController;
pub use super::integrator::bs3::BS3;
pub use super::integrator::dormand_prince::DormandPrince45; pub use super::integrator::dormand_prince::DormandPrince45;
pub use super::ode::ODE; pub use super::ode::ODE;
pub use super::problem::{Problem, Solution}; pub use super::problem::{Problem, Solution};