Numerical stability

Numerical Stability

Goal

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

Requirements

• Python 3.10+
• 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.real_part_stiffness_ratio, results.imag_dominated, results.stiff, results.recommendation, results.warning

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=1.0.

Agent workflow:

1. Check stability criteria:

   python3 scripts/cfl_checker.py --dx 0.01 --dt 1e-4 --diffusivity 1.0 --dimensions 2 --json
   

2. Interpret results:
   • Fourier number: Fo = 1.0 × 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 (the tool's recommended_dt, giving 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 not found: <path>	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 > 10⁸	Poorly-conditioned (status: poorly-conditioned)	Preconditioning likely needed
Condition number > 10¹⁰	Ill-conditioned (status: ill-conditioned)	Use scaling/preconditioning

Conditioning thresholds assume IEEE double precision: solving loses roughly log10(κ) significant digits, and a matrix becomes numerically singular near κ ≈ 1/eps ≈ 4.5e15. The > 10⁸ / > 10¹⁰ cutoffs (matching matrix_condition.py status) leave ample margin; well-conditioned-for-double FEM matrices (κ up to ~10⁶–10⁷) report status: ok.

Verification checklist

Do not declare a scheme/time step "stable" until each applicable item below is satisfied with a recorded value from the scripts, not a mental estimate.

• Ran cfl_checker.py --json and recorded metrics.cfl, metrics.fourier, and/or metrics.reaction against the reported limits.* (explicit defaults: CFL ≤ 1, Fo ≤ 1/(2d), R ≤ 1), with stable: true and the intended criteria present in criteria_applied.
• Confirmed criteria_applied is non-empty and stable is not null — i.e. at least one physics parameter (--velocity/--diffusivity/--reaction-rate) was actually supplied so a real check ran, not a silent no-op.
• Used the smallest grid spacing across all directions for --dx (anisotropic grids: smallest dx/dy/dz) and re-ran cfl_checker.py after any mesh refinement, since Fo ∝ dt/dx² makes the limit highly sensitive to dx.
• If the chosen dt is below the limit, recorded the tool's recommended_dt (and the --safety factor used) so the margin to the stability boundary is explicit and reproducible.
• For any custom/non-standard update stencil, ran von_neumann_analyzer.py --json and confirmed results.max_amplification ≤ 1 (stable: true); noted k_at_max and resolved any even-length-stencil warning.
• For multi-scale systems, ran stiffness_detector.py --json and recorded real_part_stiffness_ratio, imag_dominated, and stiff; only chose BDF/Radau when stiff: true on the real-part ratio (not on magnitude alone) and there is no warning.
• For implicit solves, ran matrix_condition.py --json and recorded condition_number and status; treated poorly-conditioned (>1e8) / ill-conditioned (>1e10) as a flag to scale/precondition before trusting the solve.

Common pitfalls & rationalizations

Tempting shortcut	Why it's wrong / what to do
"Implicit scheme, so any dt is fine."	Unconditional stability is not accuracy. cfl_checker.py reports stable: true with relaxed (infinite) limits for --scheme implicit and even adds a note to "check accuracy" — a large dt still ruins temporal error. Size dt for accuracy, not just stability.
"It ran 100 steps without crashing, so the setup is stable."	Late-time blow-up from round-off, conservation loss, or marginal Fo is common. Completion ≠ correctness — record metrics.fourier/metrics.cfl vs limits.* and confirm stable: true before trusting the run.
"The 1D Fourier limit is 0.5, so Fo ≤ 0.5 is safe."	The explicit diffusion limit is Fo ≤ 1/(2d) — 0.25 in 2D, 0.167 in 3D. Pass the real --dimensions; cfl_checker.py tightens diffusion_limit automatically, and using 0.5 in 2D/3D is an instability.
"Stiffness ratio is huge, so use BDF/Radau."	The magnitude stiffness_ratio is misleading for oscillatory/advective/Hamiltonian spectra. Check imag_dominated and real_part_stiffness_ratio: if imag_dominated: true the detector returns stiff: false with a warning — prefer symplectic/leapfrog or a CFL-sized A-stable scheme, not implicit stiff solvers.
"I tightened dt, so the old mesh's dt still works after refining dx."	Fo ∝ dt/dx²: halving dx quadruples Fo. Reusing a pre-refinement dt reintroduces a violation. Recompute with cfl_checker.py after every mesh change.
"The matrix solved, so its conditioning is fine."	A solve can return numbers while silently losing ~log10(κ) digits. Record condition_number/status from matrix_condition.py; poorly-conditioned/ill-conditioned means scale or precondition before trusting the result.

Security

Input Validation

• dx, dt, and safety are validated as finite positive numbers before any computation; velocity, diffusivity, and reaction_rate, when supplied, are validated as finite
• --dimensions is restricted to {1, 2, 3} (cfl_checker.py raises and exits 2 otherwise)
• Comma-separated eigenvalue lists (--eigs) are capped at 10,000 entries and validated as finite numbers
• Stencil coefficient lists (--coeffs) are capped at 10,000 entries and validated as finite floats

File Access

• matrix_condition.py reads a single matrix file (.npy or text) specified by --matrix; no directory traversal beyond the given path
• Matrix/Jacobian files are rejected if they exceed 500 MB before parsing (matrix_condition.py, stiffness_detector.py)
• np.load() is called with allow_pickle=False to prevent arbitrary code execution via crafted .npy files (matrix_condition.py, stiffness_detector.py)
• 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
• Bash: Used to execute the four Python analysis scripts (cfl_checker.py, von_neumann_analyzer.py, matrix_condition.py, stiffness_detector.py) with explicit argument lists
• Write: Used to save analysis results or generated reports; writes are scoped to the user's working directory
• Grep/Glob: Used to locate relevant files and search references

Safety Measures

• No eval(), exec(), or dynamic code generation
• All subprocess calls use explicit argument lists (no shell=True)
• Matrix dimension limit (100,000 per side) in matrix_condition.py prevents memory exhaustion
• JSON output mode (--json) produces structured, parseable results without shell-interpretable content

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. The stiff verdict is based on scale separation among genuinely decaying modes (Re(λ) < 0); the magnitude ratio is still reported as stiffness_ratio. Imaginary-axis-dominated spectra (oscillatory/advection/Hamiltonian) are flagged via imag_dominated and a warning, and are NOT classified as stiff — for those, prefer symplectic/leapfrog or an A-stable scheme sized by the CFL limit, not BDF/Radau.

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

• v1.2.2 (2026-06-24): Added a "Verification checklist" (7 evidence-based items tied to the four scripts' JSON outputs) and a "Common pitfalls & rationalizations" table (6 rows) to make stability verdicts reproducible and guard against false-confidence shortcuts
• v1.2.0 (2026-06-23): Corrected the phase-field worked example (genuinely unstable Fo=1.0 with D=1.0); fixed eval case 1 Fourier value; reconciled conditioning thresholds and error-message docs with the scripts; real-part-based stiffness classification with imaginary-dominated detection; enforced documented input/file-size/dimension safeguards
• v1.1.0 (2024-12-24): Enhanced documentation, decision guidance, examples
• v1.0.0: Initial release with 4 stability analysis scripts