Courant Number Calculator

Compute stable timesteps for explicit mesh simulations. View directional ratios, safety checks, exports, and plots. Make faster, safer numerical decisions with consistent grid control.

Calculator Inputs

Example Data Table

Case Model Vx Vy Vz Δt Δx Δy Δz Combined CFL Status
Advection Test A 1D 15 N/A N/A 0.010 0.250 N/A N/A 0.600 Stable
Advection Test B 2D 12 8 N/A 0.010 0.250 0.250 N/A 0.800 Near limit
Wave Test C 3D 20 10 5 0.008 0.300 0.300 0.300 0.933 Near limit

Formula Used

1D Courant Number

C = |u| × Δt / Δx

2D Conservative CFL Check

Ctotal = |u| × Δt / Δx + |v| × Δt / Δy

3D Conservative CFL Check

Ctotal = |u| × Δt / Δx + |v| × Δt / Δy + |w| × Δt / Δz

Maximum Stable Time Step

Δtmax = Limit / (|u|/Δx + |v|/Δy + |w|/Δz)

This page uses absolute velocity magnitudes and a summed directional CFL metric. That approach is conservative for many explicit transport solvers. Some discretizations use different norms or stricter limits, so always compare results with your scheme documentation.

How to Use This Calculator

  1. Choose whether your simulation is one, two, or three dimensional.
  2. Enter the velocity or characteristic wave speed for each active axis.
  3. Enter the current time step and grid spacing values.
  4. Set the stability limit. Use 1.0 unless your method requires another value.
  5. Click the calculate button to show results above the form.
  6. Review directional values, combined CFL, maximum stable Δt, and the stability message.
  7. Use the chart to see how the CFL metric changes with the time step.
  8. Download the result summary as CSV or PDF for documentation.

FAQs

1. What does the Courant number represent?

It compares how far information travels during one time step with the size of a grid cell. Larger values usually mean greater instability risk in explicit numerical methods.

2. Why do negative velocities still give positive Courant values?

Stability checks usually depend on magnitude, not sign. The sign affects transport direction, but the Courant limit is commonly evaluated using absolute speeds.

3. Is a Courant number below one always stable?

Not always. Many explicit methods use limits near one, but some higher order or multidimensional schemes require smaller values. Solver documentation should always be the final authority.

4. Why does this calculator sum directional terms?

A summed directional CFL metric is a conservative and practical check for many explicit multidimensional transport problems. It is especially useful during mesh and timestep planning.

5. What is the meaning of maximum stable Δt?

It is the largest timestep that matches your chosen limit using the entered speeds and grid spacing. Using a smaller timestep gives extra numerical safety margin.

6. Should I use cell width or node spacing for Δx?

Use the spacing consistent with your discretized governing equations. For most structured finite difference and finite volume grids, that means the local cell spacing.

7. Can I use wave speed instead of flow velocity?

Yes. For hyperbolic systems, the relevant value is the characteristic information speed. That can be an advection speed, sound speed, or another dominant propagation speed.

8. Why is an 80% safer timestep shown?

It gives a convenient buffer below the theoretical maximum. That extra margin can help when properties vary across cells or when local speeds increase during a run.

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Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.