Notes on Computational Fluid Dynamics: General Principles

3.17 Time discretisation

A local time derivative in an equation can be discretised as a finite difference in time. Time is expressed in discrete intervals, or steps, of duration .

The Euler scheme calculates the derivative from the field at the current time and the previous, or old, time level by:

(3.21)

Time discretisation thereby contributes to the diagonal (only!) coefficients of the matrix and to the source vector .

Courant number

For a 1D domain in the -direction, the Courant number is the following dimensionless parameter for each cell of length :

(3.22)

The Courant number originates from the solution of 1D advection, e.g. Eq. (2.32 ), with the Euler time scheme of Eq. (3.21 ) and an explicit upwind advection scheme from Sec. 3.10 . For that case, the Courant-Friedrichs-Lewy (CFL) condition¹² for convergence is across all cells. corresponds to a fluid particle moving across one cell in one time step, so its relevance to solution convergence is perhaps unsurprising.

In an explicit solution, the convergence limit can reduce further to or , with more accurate schemes for advection. But the finite volume method is generally implicit so stability can then be maintained with a higher maximum value of . Temporal accuracy is then the important consideration when choosing the of a simulation, both in terms of its mean and maximum value across all cells in the domain.

It is therefore important to monitor Courant number which needs to be calculated for 3D problems. Explicit, Euler, upwind discretisation of advection can be presented in 3D as follows:

(3.23)

Here are positive fluxes which transports out of the cell of interest and are negative fluxes which transports from neighbouring cells. Since the CFL condition, , requires that the coefficient of cannot be negative, it follows that

(3.24)

This 3D represents the volume of fluid leaving the cell in one time step, as a fraction of the cell volume, as shown above.

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