Methods for Unsteady Problems

In computing unsteady flows, we have a fourth coordinate direction to consider: time. Just as with the space coordinates, time must be discretized. We can consider the time “grid” in either the finite difference spirit, as discrete points in time, or in a finite volume view as “time volumes”. The major difference between the space and time coordinates lies in the direction of influence: whereas a force at any space location may (in elliptic problems) influence the flow anywhere else, forcing at a given instant will affect the flow only in the future — there is no backward influence. Unsteady flows are, therefore, parabolic-like in time. This means that no conditions can be imposed on the solution (except at the boundaries) at any time after the initiation of the calculation, which has a strong influence on the choice of solution strategy. To be faithful to the nature of time, essentially all solution methods advance in time in a step-by-step or “marching” manner. These methods are very similar to ones applied to initial value problems for ordinary differential equations (ODEs) so we shall give a brief review of such methods in the next section.