Calculator Inputs
Use positive spacings and time step. Velocity signs are converted to magnitudes for CFL evaluation.
Example Data Table
These sample cases show how changing time step, spacing, and velocities shifts both common CFL checks.
| Case | Dim. | Velocity | Δt | Δx | Δy | Δz | Summed CFL | Max CFL | Observation |
|---|---|---|---|---|---|---|---|---|---|
| 1D Shock Tube | 1D | 12.0 | 0.0004 | 0.010 | — | — | 0.48 | 0.48 | Comfortable |
| 2D Advection | 2D | 2.5, 1.5 | 0.0020 | 0.010 | 0.015 | — | 0.70 | 0.50 | Close watch |
| 2D Conservative | 2D | 1.2, 0.8 | 0.0010 | 0.010 | 0.010 | — | 0.20 | 0.12 | Comfortable |
| 3D Transport | 3D | 3.0, 1.8, 1.2 | 0.0015 | 0.010 | 0.015 | 0.020 | 0.69 | 0.45 | Manageable |
Formula Used
Directional ratios
Cx = |u| × Δt / Δx
Cy = |v| × Δt / Δy
Cz = |w| × Δt / Δz
Combined checks
Summed CFL = Cx + Cy + Cz
Maximum directional CFL = max(Cx, Cy, Cz)
Suggested largest time step
Δtmax = Climit / Σ(|velocity| / cell size) for summed mode, or Climit / max(|velocity| / cell size) for maximum mode.
The Courant number is dimensionless. Lower values generally provide more room before numerical instability appears.
How to Use This Calculator
- Select whether your case is 1D, 2D, or 3D.
- Enter the time step and the grid spacing for each active direction.
- Type the velocity components for the same directions.
- Choose whether you want a summed CFL check or a maximum directional check.
- Set the stability limit used by your solver or internal guideline.
- Press the calculate button to show the result above the form.
- Review the selected metric, directional ratios, utilization, and recommended maximum time step.
- Download the calculation as CSV or PDF for reporting.
FAQs
1. What does the Courant number measure?
It compares the distance information travels in one time step with the local grid spacing. It helps judge whether a chosen time step is numerically safe.
2. Why are there two combined CFL methods here?
Different solvers and texts use different multidimensional checks. Some sum directional contributions, while others monitor the largest directional contribution.
3. Why are velocity signs ignored?
The stability ratio depends on travel magnitude relative to grid size. Direction changes flow orientation, but the CFL magnitude usually uses absolute speeds.
4. Is a Courant number below one always stable?
Not always. Stability depends on the governing equations, spatial discretization, time integration, and boundary treatment. The limit should follow your solver guidance.
5. What does the recommended maximum time step show?
It estimates the largest time step that would meet your chosen CFL limit for the current velocities and cell sizes.
6. Should I use the summed or maximum option?
Use the method that matches your simulation documentation. If you are unsure, compare both and follow the stricter requirement.
7. Can I use this for nonuniform grids?
Yes, but enter the local cell sizes that matter for the most restrictive region. The smallest active cells usually control the allowable time step.
8. Why export CSV or PDF?
Exports make it easier to document solver settings, compare scenarios, share stability checks, and keep simulation setup records.