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