numerical-integration

Numerical Integration

Goal

Provide a reliable workflow to select integrators, set tolerances, and manage adaptive time stepping for time-dependent simulations.

Requirements

• Python 3.8+
• NumPy (for some scripts)
• No heavy dependencies for core functionality

Inputs to Gather

| Input | Description | Example |

|-------|-------------|---------|

| Problem type | ODE/PDE, stiff/non-stiff | stiff PDE |

| Jacobian available | Can compute ∂f/∂u? | yes |

| Target accuracy | Desired error level | 1e-6 |

| Constraints | Memory, implicit allowed? | implicit OK |

| Time scale | Characteristic time | 1e-3 s |

Decision Guidance

Choosing an Integrator

Is the problem stiff?
├── YES → Is Jacobian available?
│   ├── YES → Use Rosenbrock or BDF
│   └── NO → Use BDF with numerical Jacobian
└── NO → Is high accuracy needed?
    ├── YES → Use RK45 or DOP853
    └── NO → Use RK4 or Adams-Bashforth

Stiff vs Non-Stiff Detection

| Symptom | Likely Stiff | Action |

|---------|--------------|--------|

| dt shrinks to tiny values | Yes | Switch to implicit |

| Eigenvalues span many decades | Yes | Use BDF/Radau |

| Smooth solution, reasonable dt | No | Stay explicit |

Script Outputs (JSON Fields)

| Script | Key Outputs |

|--------|-------------|

| scripts/error_norm.py | error_norm, scale_min, scale_max |

| scripts/adaptive_step_controller.py | accept, dt_next, factor |

| scripts/integrator_selector.py | recommended, alternatives, notes |

| scripts/imex_split_planner.py | implicit_terms, explicit_terms, splitting_strategy |

| scripts/splitting_error_estimator.py | error_estimate, substeps |

Workflow

1. Classify stiffness - Check eigenvalue spread or use stiffness_detector
2. Choose tolerances - See references/tolerance_guidelines.md
3. Select integrator - Run scripts/integrator_selector.py
4. Compute error norms - Use scripts/error_norm.py for step acceptance
5. Adapt step size - Use scripts/adaptive_step_controller.py
6. Plan IMEX/splitting - If mixed stiff/nonstiff, use scripts/imex_split_planner.py
7. Validate convergence - Repeat with tighter tolerances

Conversational Workflow Example

User: I'm solving the Allen-Cahn equation with a stiff double-well potential. What integrator should I use?

Agent workflow:

1. Check integrator options:

   python3 scripts/integrator_selector.py --stiff --jacobian-available --accuracy high --json

1. Plan the IMEX splitting (diffusion implicit, reaction explicit):

   python3 scripts/imex_split_planner.py --stiff-terms diffusion --nonstiff-terms reaction --coupling weak --json

1. Recommend: Use IMEX-BDF2 with diffusion term implicit, double-well reaction explicit.

Pre-Integration Checklist

• [ ] Identify stiffness and dominant time scales
• [ ] Set rtol/atol consistent with physics and units
• [ ] Confirm integrator compatibility with stiffness
• [ ] Use error norm to accept/reject steps
• [ ] Verify convergence with tighter tolerance run

CLI Examples

# Select integrator for stiff problem with Jacobian
python3 scripts/integrator_selector.py --stiff --jacobian-available --accuracy high --json

# Compute scaled error norm
python3 scripts/error_norm.py --error 0.01,0.02 --solution 1.0,2.0 --rtol 1e-3 --atol 1e-6 --json

# Adaptive step control with PI controller
python3 scripts/adaptive_step_controller.py --dt 1e-2 --error-norm 0.8 --order 4 --controller pi --json

# Plan IMEX splitting
python3 scripts/imex_split_planner.py --stiff-terms diffusion,elastic --nonstiff-terms reaction --coupling strong --json

# Estimate splitting error
python3 scripts/splitting_error_estimator.py --dt 1e-4 --scheme strang --commutator-norm 50 --target-error 1e-6 --json

Error Handling

| Error | Cause | Resolution |

|-------|-------|------------|

| rtol and atol must be positive | Invalid tolerances | Use positive values |

| error-norm must be positive | Negative error norm | Check error computation |

| Unknown controller | Invalid controller type | Use i, pi, or pid |

| Splitting requires at least one term | Empty term list | Specify stiff or nonstiff terms |

Interpretation Guidance

Error Norm Values

| Error Norm | Meaning | Action |

|------------|---------|--------|

| < 1.0 | Step acceptable | Accept, maybe increase dt |

| ≈ 1.0 | At tolerance boundary | Accept with current dt |

| > 1.0 | Step rejected | Reject, reduce dt |

Controller Selection

| Controller | Properties | Best For |

|------------|------------|----------|

| I (integral) | Simple, some overshoot | Non-stiff, moderate accuracy |

| PI (proportional-integral) | Smooth, robust | General use |

| PID | Aggressive adaptation | Rapidly varying dynamics |

IMEX Strategy

| Coupling | Strategy |

|----------|----------|

| Weak | Simple operator splitting |

| Moderate | Strang splitting |

| Strong | Fully coupled IMEX-RK |

Limitations

• No automatic stiffness detection: Use stiffness_detector from numerical-stability
• Splitting assumes separability: Terms must be cleanly separable
• Jacobian requirement: Some methods need analytical or numerical Jacobian

References

• references/method_catalog.md - Integrator options and properties
• references/tolerance_guidelines.md - Choosing rtol/atol
• references/error_control.md - Error norm and adaptation formulas
• references/imex_guidelines.md - Stiff/non-stiff splitting
• references/splitting_catalog.md - Operator splitting patterns
• references/multiphase_field_patterns.md - Phase-field specific splits

Version History

• v1.1.0 (2024-12-24): Enhanced documentation, decision guidance, examples
• v1.0.0: Initial release with 5 integration scripts