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

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

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

3. 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