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 3.10+
• 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 must be a non-negative finite number / atol must be a non-negative finite number	Invalid tolerances	Use non-negative finite values
error_norm must be finite and non-negative	Negative or non-finite error norm	Check error computation
scale must be positive; with min_scale=0 ensure atol>0 or rtol*|y|>0	All scale entries collapsed to 0 (e.g. rtol=0, atol=0, default min_scale=0)	Set atol>0, rtol>0, or --min-scale > 0
argument --controller: invalid choice: ... (choose from p, pi)	Invalid controller type	Use p or pi
Provide at least one stiff or non-stiff 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	CLI value	Properties	Best For
P (elementary / integral)	p (default)	Simple, some overshoot	Non-stiff, moderate accuracy
PI (proportional-integral)	pi	Smooth, robust (requires --prev-error)	General use

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

IMEX Strategy

Coupling	Strategy
Weak	Simple operator splitting
Moderate	Strang splitting
Strong	Fully coupled IMEX-RK

Verification checklist

• Recorded the scaled error_norm (and scale_min/scale_max) from scripts/error_norm.py and confirmed scale_min > 0, so no component reduced to atol-only scaling by accident.
• Confirmed each accepted step has error_norm <= accept_threshold (default 1.0) per scripts/adaptive_step_controller.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 error_estimate, substeps, and dt_effective from scripts/splitting_error_estimator.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

Tempting shortcut	Why it's wrong / what to do
"I picked an implicit/BDF method, so any dt is fine."	Unconditional stability is not accuracy; large dt still inflates temporal error. Check error_norm against the accept threshold and run the tighter-tolerance convergence check (Workflow step 7).
"error_norm came back < 1, so the step and the whole run are correct."	The norm only certifies the local step under the chosen rtol/atol. A loose tolerance passes every step while the global solution is wrong — tighten tolerances and confirm convergence before trusting results.
"I'll use the PI controller for smoother stepping" but omit --prev-error.	Without --prev-error the script silently falls back to the P controller (controller_used: p). You get no PI benefit. Pass the previous accepted error_norm and verify controller_used: pi.
"Strang vs Lie splitting won't matter much here."	Splitting error scales as commutator_norm * dt^(order+1) with order 1 (lie) vs 2 (strang). For a nonzero commutator the schemes differ by a full power of dt — run splitting_error_estimator.py with the actual --commutator-norm and --target-error instead of guessing.
"The selector recommended IMEX/RK-Chebyshev, so I'll just run explicitly without a Jacobian."	That branch fires only for stiff problems without implicit solves and the script warns "expect smaller dt." Read the notes: provide a Jacobian/Jv product and use BDF/Radau if implicit solves are feasible.
"It ran to the final time without crashing, so the integration is valid."	Completion is not correctness. Verify step acceptance (error_norm <= threshold), splitting error within target-error, and convergence under tighter tolerances; for conservative problems pass --conservative to imex_split_planner.py and check conserved quantities.

Security

Input Validation

• All numeric inputs (dt, rtol, atol, error_norm, stiffness_ratio, commutator_norm, etc.) are validated as finite numbers at the function boundary
• imex_split_planner.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/multiphase_field_patterns.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