Numerical Integration

Select and configure time integration methods for ODE and PDE simulations — choose among explicit Runge-Kutta, BDF, Rosenbrock, and Adams families, set relative and absolute error tolerances, implement adaptive step-size control with P/PI step-size controllers, plan IMEX operator splitting for mixed stiff and non-stiff terms, and estimate splitting errors. Use when picking an integrator for a new simulation, diagnosing step rejections or tolerance failures, setting up operator splitting for phase-field or reaction-diffusion problems, or deciding between explicit and implicit time marching, even if the user only says "my solver keeps rejecting steps" or "which ODE method should I use."

HeshamFS Updated Sep 10, 2026

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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

Classify stiffness - Check eigenvalue spread or use stiffness_detector
Choose tolerances - See references/tolerance_guidelines.md
Select integrator - Run scripts/integrator_selector.py
Compute error norms - Use scripts/error_norm.py for step acceptance
Adapt step size - Use scripts/adaptive_step_controller.py
Plan IMEX/splitting - If mixed stiff/nonstiff, use scripts/imex_split_planner.py
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:

Check integrator options:

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

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

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

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

---name: numerical-integrationdescription: Select and configure time integration methods for ODE and PDE simulations — choose among explicit Runge-Kutta, BDF, Rosenbrock, and Adams families, set relative and absolute error tolerances, implement adaptive step-size control with P/PI step-size controllers, plan IMEX operator splitting for mixed stiff and non-stiff terms, and estimate splitting errors. Use when picking an integrator for a new simulation, diagnosing step rejections or tolerance failures, setting up operator splitting for phase-field or reaction-diffusion problems, or deciding between explicit and implicit time marching, even if the user only says "my solver keeps rejecting steps" or "which ODE method should I use."---# Numerical Integration## GoalProvide 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` |## Workflow1. **Classify stiffness** - Check eigenvalue spread or use stiffness_detector2. **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 acceptance5. **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: ```bash python3 scripts/integrator_selector.py --stiff --jacobian-available --accuracy high --json ```2. Plan the IMEX splitting (diffusion implicit, reaction explicit): ```bash 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```bash# Select integrator for stiff problem with Jacobianpython3 scripts/integrator_selector.py --stiff --jacobian-available --accuracy high --json# Compute scaled error normpython3 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 controllerpython3 scripts/adaptive_step_controller.py --dt 1e-2 --error-norm 0.8 --order 4 --controller pi --json# Plan IMEX splittingpython3 scripts/imex_split_planner.py --stiff-terms diffusion,elastic --nonstiff-terms reaction --coupling strong --json# Estimate splitting errorpython3 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

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