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.
Provide a repeatable checklist and script-driven checks to keep time-dependent simulations stable and defensible.
scripts/requirements.txt for dependencies| 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/sReaction rate k | First-order rate constant | 100 s⁻¹ |
| Dimensions | 1D, 2D, or 3D | 2 |
| Scheme type | Explicit or implicit | explicit |
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
| 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 | 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 |
scripts/cfl_checker.pydt if neededscripts/stiffness_detector.py if multiple scalesscripts/von_neumann_analyzer.pyscripts/matrix_condition.py for implicit solvesUser: 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:
python3 scripts/cfl_checker.py --dx 0.01 --dt 1e-4 --diffusivity 1e-3 --dimensions 2 --json
Fo = 1e-3 × 1e-4 / (0.01)² = 1.0Fo ≤ 0.252.5e-5 (to get Fo = 0.25)cfl_checker.pydt or change scheme# 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 | 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 |
| 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 |
references/stability_criteria.md - Decision thresholds and formulasreferences/common_pitfalls.md - Frequent failure modes and fixesreferences/scheme_catalog.md - Stability properties of common schemes