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numerical-integration

@HeshamFS/materials-simulation-skills
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Select and configure time integration methods for ODE/PDE simulations. Use when choosing explicit/implicit schemes, setting error tolerances, adapting time steps, diagnosing integration accuracy, planning IMEX splitting, or handling stiff/non-stiff coupled systems.

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SKILL.md

name numerical-integration
description Select and configure time integration methods for ODE/PDE simulations. Use when choosing explicit/implicit schemes, setting error tolerances, adapting time steps, diagnosing integration accuracy, planning IMEX splitting, or handling stiff/non-stiff coupled systems.
allowed-tools Read, Bash, Write, Grep, Glob

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