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:

   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

# 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