10 - PARTIAL DIFFERENTIAL EQUATIONS

Abstract

This chapter discusses the basic methods for solving partial differential equations. Partial differential equations are the basic tool for modeling physical phenomena. The amount of computer power spent in solving them is probably larger than any other single class of problems. Some of the software available is also widely applicable even though the really difficult problems almost always require programs tailored for them. Partial differential equations are similar to ordinary differential equations in so much that boundary conditions or initial values are needed to completely determine the solution. The mathematical theory of partial differential equations is quite difficult; the theory and analysis of numerical methods for partial differential equations are the most difficult and extensive in numerical computation. Not all partial differential equations are difficult to solve, and the simpler methods may be useful for implementation in a substantial number of applications. The first step in solving a partial differential equation is to discretize it; the differential equations must be replaced by an approximating, finite system of algebraic equations. A large body of numerical analysis is devoted to analyzing the error because of discretization. Once a partial differential equation has been discretized, the next step is to solve the resulting system of algebraic equations.