3.4 Discretization of the PDEs



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3.4 Discretization of the PDEs

 

For the conservative form of the transformed model equations (3.3-61) - (3.3-65), we can use finite difference expressions for the discretization. Discretizing the conservative form using finite differences is equivalent to returning to the integral form of the physical equations and to apply box integration methods for their discretization.

We find approximate representations of the differential operators which appear in the transformed equations at a given point in terms of the functions values at that point and at neighboring points. The finite difference expressions will contain function values at nine points as shown below (Figure 3.4-1). The governing physical equations do not contain mixed derivatives and therefore a five point discretization stencil were sufficient for finite differences discretization. The additional dependencies on the diagonal points are caused by the cross derivatives appearing in the transformed equations.

 

Since all the coefficients of the physical equations may be functions of the dependent variables , the discrete representation of the PDEs results in a system of nonlinear algebraic equations. This system is solved with a modified Newton algorithm. The remainder of this section deals with obtaining the discrete representation of the PDEs, the modified Newton algorithm will be explained in Section 3.5.

First we show a useful method for calculating polynomial approximations from a set of given data points in Section 3.4.1. Then we derive the discrete representations of the different terms of the PDEs in the same order as they are processed in the program code. First we obtain the expressions for the fluxes (Section 3.4.2) and the divergence (Section 3.4.3), then the recombination terms (Section 3.4.4) and time derivatives (Section 3.4.5) and finally the boundary conditions (Section 3.4.6).





Martin Stiftinger
Wed Oct 19 13:03:34 MET 1994