# Courant Number Limit#

Courant Number Limit = <float>


## Description / Usage#

This parameter’s roll is to control time step growth based on the well-known Courant number criterion. This card applies only to level-set problems. This card imposes an upper limit on the time step size, irrespective of the variable time integrator already in place.

<float>

Any floating point number to indicate the Courant number limit.

## Examples#

A sample card that sets the Courant number to 0.2 is:

Courant Number Limit = 0.2


## Technical Discussion#

See GT-034 for a thorough discussion.

## Theory#

The time step limit imposed by this limit is computed as

$\mathrm{d}t_{\mathrm{limit}} = C \min_{e} \left| \frac{h_e}{ \lVert \hat{U} \rVert_e} \right|$

Here $$e$$ is the element, $$h_e$$ is the average size of the element, $$C$$ is the specified Courant number, and

$\lVert \hat{U} \rVert_e = \frac{\int_{e}^{} \delta_{\alpha} \left( \phi \right) \underline{v} \cdot \underline{n} \, d \Omega}{\int_{e}^{} \delta_{\alpha} \left( \phi \right) \, d \Omega}$

## References#

GT-034: Tutorial on time step parameter selection for level-set problems in GOMA. April 1, 2006. D. R. Noble