differential-equations/roadmap/features/12-auto-switching.md

# 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.