TABLE OF CONTENTS We have now reached the fourth part in the succession of steps and components required to set up a CFD model, and we are, at this stage, faced with a set of space-discretized, or semi-discretized, equations.
As seen in Part III, Section 8.4, we have to make an essential decision as to the time dependence of our numerical formulation. If the physical problem is time dependent, there is obviously no choice; the mathematical initial value problem has to be discretized in time and the numerical solution has to be time accurate. On the other hand, for physical stationary problems, we can decide either to discretize the equations in space and deal with a time-independent numerical scheme or maintain the time dependency and discretize the equations in space and time, but aim only at the time asymptotic, steady, numerical solution. Remember that we have advocated that last option, as a standard discretization approach for convection-diffusion equations, such as the conservation laws.
When we select to discretize the time-dependent form, we can either develop a combined space and time discretization, such as the Lax-Wendroff schemes, as seen in the previous chapters, or perform first a separate space discretization, based on the definition of the numerical flux, leading to a semi-discretized set of ordinary differential equations (ODEs) in time.
In the next two chapters we will investigate some of the most...
