Numerical Scheme Stability Catalog
Overview
This catalog summarizes stability properties of common time integration and spatial discretization schemes used in PDE simulations.
Time Integration Schemes
Explicit Methods
| Scheme |
Order |
Stability |
CFL Limit |
Memory |
Notes |
| Forward Euler |
1 |
Conditional |
Region-dependent |
Low |
Simple but diffusive |
| RK2 (Heun) |
2 |
Conditional |
C ≤ ~1 |
Low |
Better accuracy |
| RK4 (Classic) |
4 |
Conditional |
C ≤ ~1.4 |
Low |
Most common explicit |
| RK45 (Dormand-Prince) |
4/5 |
Conditional |
Adaptive |
Medium |
Error estimation |
| Adams-Bashforth 2 |
2 |
Conditional |
C ≤ ~0.5 |
Low |
Multi-step, cheaper |
| Adams-Bashforth 4 |
4 |
Conditional |
C ≤ ~0.3 |
Low |
More restrictive |
| Leapfrog |
2 |
Neutral |
C ≤ 1 |
Low |
No damping, needs filter |
Implicit Methods
| Scheme |
Order |
Stability |
Properties |
Memory |
Notes |
| Backward Euler |
1 |
A-stable |
Unconditional |
Medium |
Very diffusive |
| Crank-Nicolson |
2 |
A-stable |
Unconditional |
Medium |
May oscillate |
| BDF2 |
2 |
A-stable |
Unconditional |
Medium |
Good for stiff |
| BDF3-6 |
3-6 |
A(α)-stable |
Nearly unconditional |
Medium |
Higher order |
| Radau IIA |
3,5 |
L-stable |
Unconditional |
High |
Very stiff problems |
| SDIRK |
2-4 |
L-stable |
Unconditional |
Medium |
Good balance |
IMEX Methods
| Scheme |
Order |
Implicit Part |
Explicit Part |
Use Case |
| IMEX-Euler |
1 |
Backward Euler |
Forward Euler |
Simple stiff/nonstiff |
| IMEX-RK2 |
2 |
SDIRK |
RK2 |
Moderate stiffness |
| ARK4(3)6L |
4 |
L-stable |
RK4 |
High-order IMEX |
Spatial Discretization Schemes
First Derivatives (Advection)
| Scheme |
Order |
Stability |
Dispersion |
Diffusion |
| Upwind (1st order) |
1 |
Stable for C ≤ 1 |
High |
Adds artificial |
| Central (2nd order) |
2 |
Unstable |
Moderate |
None |
| Upwind (3rd order) |
3 |
Stable for C ≤ 1 |
Low |
Small artificial |
| QUICK |
3 |
Conditional |
Low |
Small |
| WENO5 |
5 |
Conditional |
Very low |
Adaptive |
Second Derivatives (Diffusion)
| Scheme |
Order |
Stability (1D) |
Notes |
| Central (3-point) |
2 |
Fo ≤ 0.5 |
Standard |
| Central (5-point) |
4 |
Fo ≤ 0.5 |
Higher accuracy |
| Compact (Padé) |
4-6 |
Similar |
Implicit system |
Stability Regions
A-Stability
A method is A-stable if stable for all z = λ·dt with Re(λ) < 0.
A-stable methods:
✓ Backward Euler
✓ Crank-Nicolson
✓ BDF1-2
✓ Radau IIA
✓ All fully implicit methods
Not A-stable:
✗ All explicit methods
✗ BDF3-6 (A(α)-stable only)
✗ Adams methods
L-Stability
A method is L-stable if A-stable and |R(z)| → 0 as Re(z) → -∞.
L-stable methods:
✓ Backward Euler
✓ Radau IIA
✓ SDIRK methods
A-stable but not L-stable:
✗ Crank-Nicolson (oscillates for very stiff)
✗ Trapezoidal rule
Scheme Selection Guide
By Problem Type
| Problem Type |
Recommended Schemes |
| Smooth advection |
RK4 + Upwind/Central |
| Advection with shocks |
TVD, WENO, DG |
| Diffusion-dominated |
Implicit or Crank-Nicolson |
| Stiff reactions |
BDF, Radau, Rosenbrock |
| Wave propagation |
Leapfrog + filter, DG |
| Navier-Stokes |
IMEX, projection methods |
By Accuracy Requirement
| Accuracy |
Time Integration |
Spatial |
| Low (1st order) |
Forward/Backward Euler |
Upwind |
| Medium (2nd order) |
RK2, Crank-Nicolson |
Central, MUSCL |
| High (4th order) |
RK4, BDF4 |
Compact, DG |
| Very high (6th+) |
DOP853, Spectral |
Spectral, high-order DG |
Stability Tables
Forward Euler + Central Diffusion
| Dimensions |
Fourier Limit |
Max dt Formula |
| 1D |
Fo ≤ 0.5 |
dt ≤ dx²/(2D) |
| 2D |
Fo ≤ 0.25 |
dt ≤ dx²/(4D) |
| 3D |
Fo ≤ 0.167 |
dt ≤ dx²/(6D) |
RK4 + Central Advection
| Property |
Value |
| CFL Limit |
C ≤ 2√2 ≈ 2.83 (linear) |
| Practical Limit |
C ≤ 1.5 (nonlinear) |
BDF Methods
| Order |
Stability |
Error Constant |
| BDF1 |
A-stable, L-stable |
1 |
| BDF2 |
A-stable |
2/3 |
| BDF3 |
A(86°)-stable |
6/11 |
| BDF4 |
A(73°)-stable |
12/25 |
| BDF5 |
A(51°)-stable |
60/137 |
| BDF6 |
A(17°)-stable |
60/147 |
Comparison Matrix
| Feature |
FE |
BE |
CN |
RK4 |
BDF2 |
Radau |
| Order |
1 |
1 |
2 |
4 |
2 |
3-5 |
| A-stable |
✗ |
✓ |
✓ |
✗ |
✓ |
✓ |
| L-stable |
✗ |
✓ |
✗ |
✗ |
✗ |
✓ |
| Linear solve |
✗ |
✓ |
✓ |
✗ |
✓ |
✓ |
| Memory |
Low |
Med |
Med |
Low |
Med |
High |
| Stiff OK |
✗ |
✓ |
⚠️ |
✗ |
✓ |
✓ |
Legend: FE=Forward Euler, BE=Backward Euler, CN=Crank-Nicolson
Quick Reference
NON-STIFF: RK4, RK45, Adams-Bashforth
STIFF: BDF2, Radau, SDIRK, Rosenbrock
MIXED: IMEX schemes
WAVE PROPAGATION: Leapfrog, symplectic
SHOCK CAPTURING: TVD, WENO, DG
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