Discretization of partial differential equations

Abstract

Analytical solutions of partial differential equations involve closed-form expressionswhich give the variation of the dependent variables continuously throughoutthe domain. In contrast, numerical solutions can give answers at only discrete pointsin the domain, called grid points. For example, consider Fig. 5.1, which shows asection of a discrete grid in the xy-plane. For convenience, let us assume that thespacing of the grid points in the x-direction is uniform, and given by δx, and thatthe spacing of the points in the y-direction is also uniform, and given by δy, asshown in Fig. 5.1. In general, δx and δy are different. Indeed, it is not absolutelynecessary that δx or δy be uniform; we could deal with totally unequal spacing inboth directions, where δx is a different value between each successive pairs of gridpoints, and similarly for δy. However, the vast majority of CFD applications involvenumerical solutions on a grid which involves uniform spacing in each direction, becausethis greatly simplifies the programming of the solution, saves storage spaceand usually results in greater accuracy. This uniform spacing does not have to occurin the physical xy space; as is frequently done in CFD, the numerical calculationsare carried out in a transformed computational space which has uniform spacing inthe transformed independent variables, but which corresponds to non-uniform spacingin the physical plane. These matters will be discussed in detail in Chap. 6. In anyevent, in this chapter we will asume uniform spacing in each coordinate direction,but not necessarily equal spacing for both directions, i.e. we will assume δx and δyto be constants, but that δx does not have to equal δy.Returning to Fig. 5.1, the grid points are identified by an index i which runs in thex-direction, and an index j which runs in the y-direction. Hence, if (i, j) is the indexfor point P in Fig. 5.1, then the point immediately to the right of P is labeled as(i+1, j), the immediately to the left is (i-1, j), the point directly above is (i, j+1),and the point directly below is (i, j-1)The method of finite-differences is widely used in CFD, and therefore most ofthis chapter will be devoted to matters concerning finite differences. The philosophyof finite difference methods is to replace the partial derivatives appearing in thegoverning equations of fluid dynamics (as derived in Chap. 2) with algebraic differencequotients, yielding a system of algebraic equations which can be solved forthe flow-field variables at the specific, discrete grid points in the flow (as shown inFig. 5.1). Let us now proceed to derive some of the more common algebraic differencequotients used to discretize the partial differential equations. © Springer-Verlag Berlin Heidelberg 2009.