Stability Criteria Reference

Stability Criteria Reference

Overview

This document provides comprehensive stability criteria for explicit time integration of PDEs. Use these formulas to determine maximum stable time steps.

Core Nondimensional Numbers

Courant-Friedrichs-Lewy (CFL) Number

For advection-dominated problems:

C = v × dt / dx

Where:

• v = advection velocity (m/s)
• dt = time step (s)
• dx = grid spacing (m)

Physical meaning: Ratio of distance traveled per time step to grid spacing.

Fourier Number

For diffusion-dominated problems:

Fo = D × dt / dx²

Where:

• D = diffusivity (m²/s)
• dt = time step (s)
• dx = grid spacing (m)

Physical meaning: Ratio of diffusion distance per time step to grid spacing squared.

Reaction Number

For reaction-dominated problems:

R = k × dt

Where:

• k = reaction rate constant (1/s)
• dt = time step (s)

Physical meaning: Number of reaction time constants per time step.

Explicit Stability Limits

Advection (1D)

Advection (Multi-dimensional)

For d dimensions with velocity components v_i:

dt ≤ dx / (sum of |v_i|)    [L1 norm]
dt ≤ dx / sqrt(sum of v_i²) [L2 norm, less restrictive]

Diffusion (1D)

Diffusion (Multi-dimensional)

Fo ≤ 1 / (2 × d)

Reaction

R ≤ 1   (or R ≤ 0.2 for safety margin)

For stiff reactions (k >> 1), explicit methods are inefficient.

Combined Criteria

Advection + Diffusion

When both are present, apply both limits:

dt ≤ min(dt_advection, dt_diffusion)

dt_advection = C_limit × dx / v
dt_diffusion = Fo_limit × dx² / D

Grid Péclet Number

Ratio of advection to diffusion:

Pe = v × dx / D

Note: High Péclet numbers (Pe > 2) cause spurious oscillations with central schemes.

Advection + Diffusion + Reaction

dt ≤ min(dt_adv, dt_diff, dt_react)

Use the most restrictive limit.

Safety Factors

Recommended Safety Margins

When to Use Smaller Safety Factors

• Multiple coupled physics
• Nonlinear problems (limits may vary in time)
• Near phase transitions or sharp gradients
• Adaptive mesh refinement (dx changes)

Special Cases

Anisotropic Meshes

When dx ≠ dy ≠ dz:

dt_diff ≤ 1/(2D) × 1/(1/dx² + 1/dy² + 1/dz²)

Use the smallest spacing for the most conservative estimate.

Variable Coefficients

When v(x) or D(x) vary in space:

Use maximum values: v_max, D_max

Time-Dependent Coefficients

Re-evaluate stability at each time step or periodically.

Nonlinear Problems

Estimate local velocity/diffusivity from solution and re-check limits.

Decision Algorithm

def compute_stable_dt(dx, v, D, k, dimensions, safety=0.9):
    dt_candidates = []

    # Advection limit
    if v > 0:
        dt_adv = dx / v  # CFL = 1
        dt_candidates.append(dt_adv)

    # Diffusion limit
    if D > 0:
        fo_limit = 1.0 / (2.0 * dimensions)
        dt_diff = fo_limit * dx**2 / D
        dt_candidates.append(dt_diff)

    # Reaction limit
    if k > 0:
        dt_react = 1.0 / k
        dt_candidates.append(dt_react)

    if dt_candidates:
        return min(dt_candidates) * safety
    else:
        return None  # No limits apply

Common Mistakes

Quick Reference Card

ADVECTION:  dt ≤ dx / v           (CFL ≤ 1)
DIFFUSION:  dt ≤ dx² / (2·d·D)    (Fo ≤ 1/(2d))
REACTION:   dt ≤ 1 / k            (R ≤ 1)
COMBINED:   dt = min(all limits) × safety