# Stability

The Courant condition (neglecting viscosity effects) is used to determine the stability of an ALE process.

The maximum time step is calculated by:

$$\text{\Delta}t\le k\frac{\text{\Delta}l}{c+\nu -w}$$

Where,

- $k$
- Coefficient
- $\text{\Delta}l$
- Smallest characteristic length of an element
- $c$
- Material speed of sound
- $\nu $
- Material velocity
- $w$
- Grid velocity

The speed of sound is determined by:

$$c=\sqrt{\frac{1}{\rho}\frac{\partial p}{\partial \rho}+\frac{4}{3}\frac{\mu}{\rho}}$$

Where,

- $\rho $
- Density
- $\mu $
- Dynamic viscosity
- $p$
- Pressure

The relative velocity between the material and grid motion ( $\nu $ - $w$ ) is computed by:

$$\nu -w=\sqrt{\frac{1}{N}{{\displaystyle \sum _{i=1}^{3}{\displaystyle \sum _{I=1}^{N}\left({\nu}_{i}^{I}-{w}_{i}^{I}\right)}}}^{2}}$$

Where,

- $N$
- Number of nodes of the considered element (usually $N$ =8)