Numerical Stability

---
author: luo-kai
name: numerical-stability
description: Analyze and enforce numerical stability for time-dependent PDE simulations. Use when selecting time steps, choosing explicit/implicit schemes, diagnosing numerical blow-up, checking CFL/Fourier criteria, von Neumann analysis, matrix conditioning, or detecting stiffness in advection/diffusion/reaction problems.
allowed-tools: Read, Bash, Write, Grep, Glob
---

# Numerical Stability

## Goal

Provide a repeatable checklist and script-driven checks to keep time-dependent simulations stable and defensible.

## Requirements

- Python 3.8+
- NumPy (for matrix_condition.py and von_neumann_analyzer.py)
- See `scripts/requirements.txt` for dependencies

## Inputs to Gather

| Input | Description | Example |
|-------|-------------|---------|
| Grid spacing `dx` | Spatial discretization | `0.01 m` |
| Time step `dt` | Temporal discretization | `1e-4 s` |
| Velocity `v` | Advection speed | `1.0 m/s` |
| Diffusivity `D` | Thermal/mass diffusivity | `1e-5 m²/s` |
| Reaction rate `k` | First-order rate constant | `100 s⁻¹` |
| Dimensions | 1D, 2D, or 3D | `2` |
| Scheme type | Explicit or implicit | `explicit` |

## Decision Guidance

### Choosing Explicit vs Implicit

```
Is the problem stiff (fast + slow dynamics)?
├── YES → Use implicit or IMEX scheme
│         └── Check conditioning with matrix_condition.py
└── NO → Is CFL/Fourier satisfied with reasonable dt?
    ├── YES → Use explicit scheme (cheaper per step)
    └── NO → Consider implicit or reduce dx
```

### Stability Limit Quick Reference

| Physics | Number | Explicit Limit (1D) | Formula |
|---------|--------|---------------------|---------|
| Advection | CFL | C ≤ 1 | `C = v·dt/dx` |
| Diffusion | Fourier | Fo ≤ 0.5 | `Fo = D·dt/dx²` |
| Reaction | Reaction | R ≤ 1 | `R = k·dt` |

**Multi-dimensional correction**: For d dimensions, diffusion limit is `Fo ≤ 1/(2d)`.

## Script Outputs (JSON Fields)

| Script | Key Outputs |
|--------|-------------|
| `scripts/cfl_checker.py` | `metrics.cfl`, `metrics.fourier`, `recommended_dt`, `stable` |
| `scripts/von_neumann_analyzer.py` | `results.max_amplification`, `results.stable` |
| `scripts/matrix_condition.py` | `results.condition_number`, `results.is_symmetric` |
| `scripts/stiffness_detector.py` | `results.stiffness_ratio`, `results.stiff`, `results.recommendation` |

## Workflow

1. **Identify dominant physics** (advection vs diffusion vs reaction)
2. **Run CFL checker** with `scripts/cfl_checker.py`
3. **Compare to limits** and adjust `dt` if needed
4. **Check stiffness** with `scripts/stiffness_detector.py` if multiple scales
5. **Analyze custom schemes** with `scripts/von_neumann_analyzer.py`
6. **Check conditioning** with `scripts/matrix_condition.py` for implicit solves
7. **Document** the stability verdict and recommended time step

## Conversational Workflow Example

**User**: My phase-field simulation is blowing up after 100 steps. I'm using explicit Euler with dx=0.01, dt=1e-4, and diffusivity D=1e-3.

**Agent workflow**:
1. Check stability criteria:
   ```bash
   python3 scripts/cfl_checker.py --dx 0.01 --dt 1e-4 --diffusivity 1e-3 --dimensions 2 --json
   ```
2. Interpret results:
   - Fourier number: `Fo = 1e-3 × 1e-4 / (0.01)² = 1.0`
   - 2D limit: `Fo ≤ 0.25`
   - **Violation**: Fo = 1.0 > 0.25, unstable!
3. Recommend fix:
   - Reduce dt to `2.5e-5` (to get Fo = 0.25)
   - Or increase dx, or switch to implicit

## Pre-Simulation Stability Checklist

- [ ] Identify dominant physics and nondimensional groups
- [ ] Compute CFL/Fourier/Reaction numbers with `cfl_checker.py`
- [ ] If explicit and limit violated, reduce `dt` or change scheme
- [ ] If stiffness ratio > 1000, select implicit/stiff integrator
- [ ] For custom schemes, verify amplification factor ≤ 1
- [ ] Document stability reasoning with inputs and outputs

## CLI Examples

```bash
# Check CFL/Fourier for 2D diffusion-advection
python3 scripts/cfl_checker.py --dx 0.1 --dt 0.01 --velocity 1.0 --diffusivity 0.1 --dimensions 2 --json

# Von Neumann analysis for custom 3-point stencil
python3 scripts/von_neumann_analyzer.py --coeffs 0.2,0.6,0.2 --dx 1.0 --nk 128 --json

# Detect stiffness from eigenvalue estimates
python3 scripts/stiffness_detector.py --eigs=-1,-1000 --json

# Check matrix conditioning for implicit system
python3 scripts/matrix_condition.py --matrix A.npy --norm 2 --json
```

## Error Handling

| Error | Cause | Resolution |
|-------|-------|------------|
| `dx and dt must be positive` | Zero or negative values | Provide valid positive numbers |
| `No stability criteria applied` | Missing velocity/diffusivity | Provide at least one physics parameter |
| `Matrix file not found` | Invalid path | Check matrix file exists |
| `Could not compute eigenvalues` | Singular or ill-formed matrix | Check matrix validity |

## Interpretation Guidance

| Scenario | Meaning | Action |
|----------|---------|--------|
| `stable: true` | All checked criteria satisfied | Proceed with simulation |
| `stable: false` | At least one limit violated | Reduce dt or change scheme |
| `stable: null` | No criteria could be applied | Provide more physics inputs |
| Stiffness ratio > 1000 | Problem is stiff | Use implicit integrator |
| Condition number > 10⁶ | Ill-conditioned | Use scaling/preconditioning |

## Limitations

- **Explicit schemes only** for CFL/Fourier checks (implicit is unconditionally stable)
- **Von Neumann analysis** assumes linear, constant-coefficient, periodic BCs
- **Stiffness detection** requires eigenvalue estimates from user

## References

- `references/stability_criteria.md` - Decision thresholds and formulas
- `references/common_pitfalls.md` - Frequent failure modes and fixes
- `references/scheme_catalog.md` - Stability properties of common schemes

## Version History

- **v1.1.0** (2024-12-24): Enhanced documentation, decision guidance, examples
- **v1.0.0**: Initial release with 4 stability analysis scripts