OpenFOAM Courant Number Too High: Causes and Solutions

The Courant number (Co) is the ratio of the distance a fluid particle travels in one time step to the size of a mesh cell. When Co exceeds 1, information propagates faster than the mesh can resolve it — and the simulation diverges. Understanding and controlling Co is essential for any transient OpenFOAM simulation.

By CFDpilot · Updated April 2026

What the Courant number means

The CFL (Courant-Friedrichs-Lewy) condition states:

Co = U · Δt / Δx

Where U is the local fluid velocity, Δt is the time step, and Δx is the local cell size. For an explicit time-marching scheme, numerical stability requires Co < 1 in every cell.

In the OpenFOAM log you'll see:

Courant Number mean: 0.127  max: 0.843

The max value is what matters for stability. It occurs in the smallest cells or highest-velocity regions — typically near wall boundaries or in refined regions of the mesh.

Why Co > 1 causes divergence

When Co > 1, a fluid particle travels more than one cell width per time step. This means the upstream information that should influence the current cell is not accounted for — the scheme is trying to extrapolate beyond its stencil. For explicit schemes (like the pure PISO algorithm in icoFoam), this produces velocities that violate physical bounds and grow without limit.

For implicit schemes (like PIMPLE with multiple outer correctors), the coupling between pressure and velocity allows larger Co. However, accuracy degrades as Co increases even if the simulation remains stable — temporal discretisation error grows proportionally to Co above 1.

How to calculate the right deltaT

Before running, estimate the maximum velocity in your domain and the minimum cell size:

// Co_target = 0.8 (conservative)
// Δt = Co_target * Δx_min / U_max
//
// Example: U_max = 5 m/s, Δx_min = 0.002 m (snappyHexMesh refinement)
// Δt = 0.8 * 0.002 / 5 = 3.2e-4 s

Find Δx_min from the checkMesh output (look for Minimum face area or estimate from your refinement levels).

Estimating deltaT from snappyHexMesh refinement levels

If you know your base mesh cell size and refinement levels, you can estimate the minimum cell size directly. Each refinement level halves the cell size:

// Base mesh: 0.1 m cells (blockMesh)
// Refinement level 3: 0.1 / 2^3 = 0.0125 m
// Refinement level 5: 0.1 / 2^5 = 0.003125 m
//
// U_max = 20 m/s, Co_target = 0.5
// Δt = 0.5 * 0.003125 / 20 = 7.8e-5 s ≈ 1e-4 s (rounded down for safety)

Always round deltaT down rather than up when setting it manually. The cost of halving deltaT is doubled computation time — but the cost of a diverged simulation is starting over entirely.

Using adjustable time stepping

The best practice for transient simulations is to let OpenFOAM automatically adjust deltaT to maintain a target Courant number. Configure in controlDict:

application     pimpleFoam;

startFrom       startTime;
startTime       0;
stopAt          endTime;
endTime         1.0;

deltaT          1e-4;       // initial deltaT
adjustTimeStep  yes;
maxCo           0.8;       // target max Courant number
maxDeltaT       0.01;      // upper bound

writeControl    adjustableRunTime;
writeInterval   0.1;

With adjustable time stepping, OpenFOAM increases deltaT when Co is low (saving computation time) and reduces it when Co approaches the limit (maintaining stability).

How the adjustable time step algorithm works

At each time step, pimpleFoam computes the current maximum Courant number Co_max. It then scales deltaT for the next step by the factor maxCo / Co_max: if Co_max is 0.4 and maxCo is 0.8, deltaT doubles. If Co_max is 0.9 and maxCo is 0.8, deltaT is reduced by 11%. The maxDeltaT parameter prevents deltaT from growing too large during quiet flow periods. Use writeControl adjustableRunTime to ensure outputs occur at regular physical time intervals despite varying deltaT.

Solver-specific Co limits

The acceptable Courant number depends on the solver and time integration scheme:

• icoFoam, pisoFoam — Co < 1 strictly required (first-order Euler)
• pimpleFoam with 1 outer corrector — Co < 1 recommended
• pimpleFoam with 2–3 outer correctors — Co up to 5–10 possible (implicit scheme benefit)
• interFoam (VOF) — Co < 0.5 for the interface region; use maxAlphaCo
• rhoCentralFoam — acoustic CFL must also be satisfied: Co < 1 based on speed of sound

Configuring pimpleFoam outer correctors for large Co

When using pimpleFoam with multiple outer correctors, configure the PIMPLE block in fvSolution:

PIMPLE
{
    nOuterCorrectors    3;   // allows Co up to ~5-10
    nCorrectors         2;   // inner pressure correctors
    nNonOrthogonalCorrectors 1;
    turbOnFinalIterationOnly yes;  // efficiency: update turbulence last
}

Each outer corrector performs a full PISO loop. Three outer correctors allow larger deltaT but increase computational cost per time step — the trade-off is worthwhile when the required physical simulation time is long relative to the Courant-limited deltaT.

Courant number in steady-state simulations

simpleFoam and other steady-state solvers do not use time stepping in the physical sense — they use pseudo-time iterations. The Courant number displayed in the log for these solvers is a pseudo-Co and does not directly control stability. For steady-state simulations, the Courant number is not the root cause of divergence — look at relaxation factors and discretisation schemes instead.

A very high pseudo-Co (e.g. 5000) in a simpleFoam log is often a symptom of divergence, not its cause.

interFoam: Courant number for the interface

For VOF simulations with interFoam, there are two Courant numbers to watch: the standard Co and the interface Co (maxAlphaCo). The interface requires a tighter limit to prevent smearing:

adjustTimeStep  yes;
maxCo           1.0;
maxAlphaCo      0.5;   // interface Courant limit
maxDeltaT       1e-3;

Why the interface needs a lower Co limit

In VOF methods, the interface is represented by the gradient of the alpha field. The MULES (Multidimensional Limiter for Explicit Solutions) algorithm used by interFoam to advect alpha has its own stability constraint, independent of the velocity field stability. For sharp interfaces, maxAlphaCo 0.5 is required to prevent the interface from smearing over multiple cells within a single time step. When interFoam output shows Interface Courant Number mean: X max: Y, the alpha-Co constrained by maxAlphaCo overrides the bulk Co constraint when it is the binding limit.

Diagnosing Courant number problems from the log

When the simulation is approaching instability due to high Co, the log shows a characteristic pattern before the crash:

Time = 0.0018
Courant Number mean: 0.543  max: 0.872
...
Time = 0.0019
Courant Number mean: 1.23   max: 3.45
...
Time = 0.0019
Courant Number mean: 45.2   max: 187.3
Floating point exception (core dumped)

The jump from Co=0.87 to Co=3.45 in one step indicates a sudden velocity spike in one or a few cells — the beginning of the CFL instability cascade. Within a few more steps, NaN propagates through the domain. The fix is to either reduce deltaT so the initial spike never happens, or use adjustable time stepping so the solver automatically reduces deltaT when Co climbs toward the limit.

rhoCentralFoam: acoustic Courant number

For compressible flows with rhoCentralFoam, the CFL condition must be satisfied for both convective and acoustic waves. The acoustic Courant number is based on the speed of sound rather than the flow velocity:

// Acoustic Co = (U + c) * deltaT / deltaX
// c = speed of sound = sqrt(gamma * R * T)
// For air at 300K: c = sqrt(1.4 * 287 * 300) = 347 m/s
//
// Example: U=50 m/s, c=347 m/s, deltaX=0.01m, deltaT=1e-5s
// Acoustic Co = (50 + 347) * 1e-5 / 0.01 = 0.397 — OK

In supersonic flows (Mach > 1), the acoustic Courant number requirement dominates and forces very small deltaT, making compressible simulations significantly more expensive than incompressible ones at similar Reynolds numbers.