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

1. 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!

1. 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