23 - Finite Volume Methods on Quadrilateral Grids

Summary

Many multidimensional problems of practical interest involve complex geometry, and in general it is not sufficient to be able to solve hyperbolic equations on a uniform Cartesian grid in a rectangular domain. In Section 6.17 we considered a nonuniform grid in one space dimension and sawhowhyperbolic equations can be solved on such a grid by using a uniform grid in computational space together with a coordinate mapping and appropriate scaling of the flux differences using capacity form differencing. The capacity of the computational cell is determined by the size of the corresponding physical cell.

In this chapter we consider nonuniform finite volume grids in two dimensions, such as those shown in Figure 23.1, and will see that similar techniques may be used. There are various ways to view the derivation of finite volume methods on general multidimensional grids. Here we will consider a direct physical interpretation in terms of fluxes normal to the cell edges. For simplicity we restrict attention to two space dimensions. For some other discussions of finite volume methods on general grids, see for example.

The grids shown in Figures 23.1(a) and (b) are logically rectangular quadrilateral grids, and we will concentrate on this case. Each cell is a quadrilateral bounded by four linear segments. Such a grid is also often called a curvilinear grid.