Numerical Integration

Goal

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

Requirements

• Python 3.10+
• NumPy (for some scripts)
• No heavy dependencies for core functionality

Inputs to Gather

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

Script Outputs (JSON Fields)

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/adaptivestepcontroller.py
6. Plan IMEX/splitting - If mixed stiff/nonstiff, use scripts/imexsplitplanner.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:

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

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

``bash python3 scripts/imexsplitplanner.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

Interpretation Guidance

Error Norm Values

Controller Selection

PID control is described in references/error_control.md for reference only; the
adaptivestepcontroller.py CLI implements p and pi controllers.

IMEX Strategy

Verification checklist

• Recorded the scaled errornorm (and scalemin/scalemax) from scripts/errornorm.py and confirmed scale_min > 0, so no component reduced to atol-only scaling by accident.
• Confirmed each accepted step has errornorm <= acceptthreshold (default 1.0) per scripts/adaptivestepcontroller.py; logged any accept: false steps and the resulting dt_next/factor rather than forcing the step through.
• For the PI controller, supplied --prev-error and verified controller_used reported pi (not the silent p fallback that occurs when --prev-error is omitted).
• Ran scripts/integrator_selector.py with the actual --stiff/--jacobian-available/--accuracy flags matching the problem and recorded recommended plus the notes (e.g. the "expect smaller dt" warning for stiff-without-implicit).
• For operator splitting, recorded errorestimate, substeps, and dteffective from scripts/splittingerrorestimator.py and confirmed error_estimate <= target-error after substepping, using the correct --scheme order (lie=1, strang=2).
• Ran the convergence validation (Workflow step 7): repeated with tighter rtol/atol and confirmed the solution change is below the target accuracy, rather than trusting a single tolerance run.

Common pitfalls & rationalizations

Security

Input Validation

• All numeric inputs (dt, rtol, atol, errornorm, stiffnessratio, commutator_norm, etc.) are validated as finite numbers at the function boundary
• imexsplitplanner.py validates term names against [a-zA-Z][a-zA-Z0-9 -]* with length and count limits, preventing injection payloads in user-supplied term lists
• Comma-separated value lists are capped at 100,000 entries to prevent resource exhaustion
• Numeric bounds enforced: dimension capped at 10 billion, order at 20, stiffness_ratio at 1e30
• --controller is validated against a fixed allowlist (p, pi)
• --scheme is validated against known splitting schemes (lie, strang)

File Access

• Scripts read no external files; all inputs are provided via CLI arguments
• Scripts write only to stdout (JSON output); no files are created unless the agent explicitly uses the Write tool

Tool Restrictions

• Read: Used to inspect script source, references, and user configuration files
• Write: Used to save integrator recommendations or splitting plans; writes are scoped to the user's working directory
• Grep/Glob: Used to locate relevant files and search references
• The skill's allowed-tools excludes Bash to prevent the agent from executing arbitrary commands when processing user-provided inputs

Safety Measures

• No eval(), exec(), or dynamic code generation
• All subprocess calls use explicit argument lists (no shell=True)
• Reduced tool surface (no Bash) limits the agent to read/write operations only
• Term names are sanitized before use, preventing shell metacharacter injection

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/multiphasefieldpatterns.md - Phase-field specific splits

Version History

• v1.2.2 (2026-06-24): Added Verification checklist and Common pitfalls & rationalizations sections grounding step acceptance, PI controller usage, integrator selection, and splitting-error checks in the scripts' actual outputs
• v1.2.0 (2026-06-23): Fixed PI controller coefficient/sign to standard form, fixed error_norm scale-collapse crash, corrected error-message and controller documentation to match the scripts
• v1.1.0 (2024-12-24): Enhanced documentation, decision guidance, examples
• v1.0.0: Initial release with 5 integration scripts