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