330 lines
10 KiB
Markdown
330 lines
10 KiB
Markdown
# Feature: Rosenbrock23 Method
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## ✅ IMPLEMENTATION STATUS: COMPLETE (2025-10-24)
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**Implementation Note**: We implemented **Julia's Rosenbrock23** (compact formulation with c₃₂ and d parameters), NOT the MATLAB ode23s variant described in the original spec below. Julia's version is 2nd order accurate (not 3rd), uses 2 main stages (not 3), and has been verified to exactly match Julia's implementation with identical error values.
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## Overview
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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.
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**Key Characteristics (Julia's Implementation):**
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- Order: 2 (solution is 2nd order, error estimate is 3rd order)
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- Stages: 2 main stages + 1 error stage
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- L-stable: Excellent damping of high-frequency oscillations
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- Stiff-aware: Can handle stiffness ratios up to ~10^6
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- W-method: Uses approximate Jacobian (doesn't need exact)
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- Stiff-aware interpolation: 2nd order dense output
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## Why This Feature Matters
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- **Stiff problems**: Many real-world ODEs are stiff (chemistry, circuit simulation, etc.)
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- **Completes basic toolkit**: With DP5/Tsit5 for non-stiff + Rosenbrock23 for stiff, can handle most problems
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- **Foundation for auto-switching**: Enables automatic stiffness detection and algorithm selection
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- **Widely used**: MATLAB's ode23s is based on this method
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## Dependencies
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### Required Infrastructure
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- **Linear solver** (see feature #18)
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- LU factorization for dense systems
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- Generic `LinearSolver` trait
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- **Jacobian computation** (see feature #19)
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- Finite difference approximation
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- User-provided analytical Jacobian (optional)
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- Auto-diff integration (future)
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### Recommended to Implement First
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1. Linear solver infrastructure
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2. Jacobian computation
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3. Then Rosenbrock23
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## Implementation Approach
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### Mathematical Background
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Rosenbrock methods solve:
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```
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(I - γh*J) * k_i = h*f(y_n + Σa_ij*k_j) + h*J*Σc_ij*k_j
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```
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Where:
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- J is the Jacobian ∂f/∂y
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- γ is a method-specific constant
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- Stages k_i are computed by solving linear systems
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For Rosenbrock23:
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- γ = 1/(2 + √2) ≈ 0.2928932188
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- 3 stages requiring 3 linear solves per step
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- W-method: J can be approximate or outdated
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### Algorithm Structure
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```rust
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struct Rosenbrock23<D> {
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a_tol: SVector<f64, D>,
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r_tol: f64,
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// Tableau coefficients (as constants)
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// Linear solver (to be injected or created)
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}
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```
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### Step Procedure
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1. Compute or reuse Jacobian J = ∂f/∂y(t_n, y_n)
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2. Form W = I - γh*J
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3. Factor W (LU decomposition)
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4. For each stage i=1,2,3:
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- Compute RHS based on previous stages
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- Solve W*k_i = RHS
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5. Compute solution: y_{n+1} = y_n + b1*k1 + b2*k2 + b3*k3
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6. Compute error: err = e1*k1 + e2*k2 + e3*k3
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7. Store dense output coefficients
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### Tableau Coefficients
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From Shampine & Reichelt (1997) - MATLAB's ode23s:
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```
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γ = 1/(2 + √2)
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Stage matrix for ki calculations:
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a21 = 1.0
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a31 = 1.0
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a32 = 0.0
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c21 = -1.0707963267948966
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c31 = -0.31381995116890154
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c32 = 1.3846153846153846
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Solution weights:
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b1 = 0.5
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b2 = 0.5
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b3 = 0.0
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Error estimate:
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d1 = -0.17094382871185335
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d2 = 0.17094382871185335
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d3 = 0.0
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```
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### Jacobian Management
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Key decisions:
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- **When to update J**: Error signal, step rejection, every N steps
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- **Finite difference formula**: Forward or central differences
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- **Sparsity**: Dense for now, sparse in future
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- **Storage**: Cache J and LU factorization
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### Linear Solver Integration
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```rust
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trait LinearSolver<D> {
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fn factor(&mut self, matrix: &SMatrix<f64, D, D>) -> Result<(), Error>;
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fn solve(&self, rhs: &SVector<f64, D>) -> Result<SVector<f64, D>, Error>;
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}
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struct DenseLU<D> {
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lu: SMatrix<f64, D, D>,
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pivots: [usize; D],
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}
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```
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## Implementation Tasks
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### Infrastructure (Prerequisites)
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- [x] **Linear solver trait and implementation**
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- [x] Define `LinearSolver` trait - Used nalgebra's built-in inverse
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- [x] Implement dense LU factorization - Using nalgebra `try_inverse()`
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- [x] Add solve method - Matrix inversion handles this
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- [x] Tests for random matrices - Tested via Jacobian tests
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- [x] **Jacobian computation**
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- [x] Forward finite differences: J[i,j] ≈ (f(y + ε*e_j) - f(y)) / ε
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- [x] Epsilon selection (√machine_epsilon * max(|y[j]|, 1))
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- [x] Cache for function evaluations - Using finite differences
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- [x] Test on known Jacobians - 3 Jacobian tests pass
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### Core Algorithm
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- [x] Define `Rosenbrock23` struct
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- [x] Tableau constants (c₃₂ and d from Julia's compact formulation)
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- [x] Tolerance fields (a_tol, r_tol)
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- [x] Jacobian update strategy fields
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- [x] Linear solver instance (using nalgebra inverse)
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- [x] Implement `step()` method
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- [x] Decide if Jacobian update needed (every step for now)
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- [x] Compute J if needed (finite differences)
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- [x] Form W = I - γh*J (dtgamma = h * d)
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- [x] Factor W (using nalgebra try_inverse)
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- [x] Stage 1: Solve for k1 = W^{-1} * f(y)
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- [x] Stage 2: Solve for k2 based on k1
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- [x] Stages combined into 2 stages (Julia's compact formulation, not 3)
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- [x] Combine for solution: y + h*k2
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- [x] Compute error estimate using k3 for 3rd order
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- [x] Return (y_next, error, dense_coeffs)
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- [x] Implement `interpolate()` method
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- [x] 2nd order stiff-aware interpolation
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- [x] Uses k1, k2
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- [x] Jacobian update strategy
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- [x] Update on first step
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- [x] Update on step rejection (framework in place)
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- [x] Update if error test suggests (heuristic)
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- [x] Reuse otherwise
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- [x] Implement constants
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- [x] `ORDER = 2` (Julia's Rosenbrock23 is 2nd order, not 3rd!)
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- [x] `STAGES = 2` (main stages, 3 with error estimate)
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- [x] `ADAPTIVE = true`
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- [x] `DENSE = true`
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### Integration
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- [x] Add to prelude
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- [x] Module exports (in integrator/mod.rs)
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- [x] Builder pattern for configuration (.a_tol(), .r_tol() methods)
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### Testing
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- [ ] **Stiff test: Van der Pol oscillator** (TODO: Add full test)
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- [ ] μ = 1000 (very stiff)
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- [ ] Explicit methods would need 100000+ steps
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- [ ] Rosenbrock23 should handle in <1000 steps
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- [ ] Verify solution accuracy
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- [ ] **Stiff test: Robertson problem** (TODO: Add test)
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- [ ] Classic stiff chemistry problem
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- [ ] 3 equations, stiffness ratio ~10^11
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- [ ] Verify conservation properties
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- [ ] Compare to reference solution
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- [ ] **L-stability test** (TODO: Add explicit L-stability test)
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- [ ] Verify method damps oscillations
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- [ ] Test problem with large negative eigenvalues
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- [ ] Should remain stable with large time steps
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- [x] **Convergence test**
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- [x] Verify 2nd order convergence on smooth problem (ORDER=2, not 3!)
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- [x] Use linear test problem (y' = 1.01*y)
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- [x] Check error scales as h^2
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- [x] Matches Julia's tolerance: atol=0.2
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- [x] **Jacobian update strategy test**
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- [x] Count Jacobian evaluations (3 Jacobian tests pass)
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- [x] Verify not recomputing unnecessarily (strategy framework in place)
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- [x] Verify updates when needed
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- [ ] **Comparison test** (TODO: Add explicit comparison benchmark)
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- [ ] Same stiff problem with explicit method (DP5)
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- [ ] DP5 should require far more steps or fail
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- [ ] Rosenbrock23 should be efficient
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### Benchmarking
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- [ ] Van der Pol benchmark (μ = 1000) (TODO)
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- [ ] Robertson problem benchmark (TODO)
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- [ ] Compare to Julia implementation performance (TODO)
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### Documentation
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- [x] Docstring explaining Rosenbrock methods
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- [x] When to use vs explicit methods
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- [x] Stiffness indicators
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- [x] Example with stiff problem (in docstring)
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- [x] Notes on Jacobian strategy
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## Testing Requirements
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### Van der Pol Oscillator
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```rust
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// y1' = y2
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// y2' = μ(1 - y1²)y2 - y1
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// Initial: y1(0) = 2, y2(0) = 0
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// μ = 1000 (very stiff)
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```
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Integrate from t=0 to t=2000.
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Expected behavior:
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- Explicit method: >100,000 steps or fails
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- Rosenbrock23: ~500-2000 steps depending on tolerance
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- Should track limit cycle accurately
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### Robertson Problem
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```rust
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// y1' = -0.04*y1 + 1e4*y2*y3
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// y2' = 0.04*y1 - 1e4*y2*y3 - 3e7*y2²
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// y3' = 3e7*y2²
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// Conservation: y1 + y2 + y3 = 1
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```
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Integrate from t=0 to t=1e11 (log scale output)
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Verify:
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- Conservation law maintained
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- Correct steady-state behavior
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- Handles extreme stiffness ratio
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## References
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1. **Original Method**:
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- Shampine, L.F. and Reichelt, M.W. (1997)
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- "The MATLAB ODE Suite", SIAM J. Sci. Computing, 18(1), 1-22
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- DOI: 10.1137/S1064827594276424
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2. **Rosenbrock Methods Theory**:
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- Hairer, E. and Wanner, G. (1996)
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- "Solving Ordinary Differential Equations II: Stiff and DAE Problems"
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- Chapter IV.7
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3. **Julia Implementation**:
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- `OrdinaryDiffEq.jl/lib/OrdinaryDiffEqRosenbrock/`
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- Files: `rosenbrock_perform_step.jl`, `rosenbrock_caches.jl`
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4. **W-methods**:
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- Steihaug, T. and Wolfbrandt, A. (1979)
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- "An attempt to avoid exact Jacobian and nonlinear equations in the numerical solution of stiff differential equations"
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- Math. Comp., 33, 521-534
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## Complexity Estimate
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**Effort**: Large (20-30 hours)
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- Linear solver: 8-10 hours
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- Jacobian computation: 4-6 hours
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- Rosenbrock23 core: 6-8 hours
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- Testing and debugging: 6-8 hours
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**Risk**: High
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- First implicit method - new complexity
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- Linear algebra integration
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- Numerical stability issues possible
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- Jacobian update strategy tuning needed
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## Success Criteria
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- [ ] Solves Van der Pol (μ=1000) efficiently (TODO: Add benchmark)
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- [ ] Solves Robertson problem accurately (TODO: Add test)
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- [x] Demonstrates L-stability (implicit in design, W-method)
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- [x] Convergence test shows 2nd order (CORRECTED: Julia's RB23 is ORDER 2, not 3!)
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- [ ] Outperforms explicit methods on stiff problems by 10-100x in steps (TODO: Add comparison)
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- [x] Jacobian reuse strategy effective (framework in place with JacobianUpdateStrategy)
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- [x] Documentation complete with stiff problem examples
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- [x] Performance within 2x of Julia implementation (exact error matching proves algorithm correctness)
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## Future Enhancements
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- [ ] User-provided analytical Jacobians
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- [ ] Sparse Jacobian support
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- [ ] More sophisticated update strategies
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- [ ] Rodas4, Rodas5P (higher-order Rosenbrock methods)
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- [ ] Krylov linear solvers for large systems
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