Finite-Difference Approximations

Abstract

In common with the equations governing unsteady fluid flow, our model equations contain partial derivatives with respect to both space and time. One can approximate these simultaneously and then solve the resulting difference equations. Alternatively, one can approximate the spatial derivatives first, thereby producing a system of ordinary differential equations. The time derivatives are approximated next, leading to a time-marching method which produces a set of difference equations. This is the approach emphasized here. In this chapter, the concept of finite-difference approximations to partial derivatives is presented. While these can be applied either to spatial derivatives or time derivatives, our emphasis in this chapter is on spatial derivatives; time derivatives are treated in Chapter 6. Strategies for applying these finite-difference approximations will be discussed in Chapter 4.