3.17 Time discretisation

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

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

@----! -------o @t t \relax \special {t4ht=

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

Courant number

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

Co = ux---t: x \relax \special {t4ht=
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 eqn across all cells. eqn 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 eqn or eqn, 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 eqn. Temporal accuracy is then the important consideration when choosing the eqn of a simulation, both in terms of its mean and maximum value across all cells in the domain.

PICT\relax \special {t4ht=

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:

 X X = o --t + o -t- o = 0: V f V f N \relax \special {t4ht=
Here eqn are positive fluxes which transports eqn out of the cell of interest and eqn are negative fluxes which transports eqn from neighbouring cells. Since the CFL condition, eqn, requires that the coefficient of eqn cannot be negative, it follows that
 --tX + Co = V : f \relax \special {t4ht=
This 3D eqn represents the volume of fluid leaving the cell in one time step, as a fraction of the cell volume, as shown above.
12after Richard Courant, Kurt Friedrichs and Hans Lewy, Uber die partiellen Differenzengleichungen der mathematischen Physik, 1928.

Notes on CFD: General Principles - 3.17 Time discretisation