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**Unformatted text preview: **Chapter 5 Numerical Methods: Finite Differences As you know, the differential equations that can be solved by an explicit analytic formula are few and far between. Consequently, the development of accurate numeri- cal approximation schemes is essential for extracting quantitative information as well as achieving a qualitative understanding of the behavior of their solutions. Even in cases, such as the heat and wave equations, where explicit solution formulas (either closed form or infinite series) exist, numerical methods can still be profitably employed. Indeed, one can accurately test a proposed numerical algorithm by running it on a known solution. Fur- thermore, the lessons learned in the design of numerical algorithms for solved examples are of inestimable value when confronting more challenging problems. Many basic numerical solution schemes for partial differential equations can be fit into two broad themes. The first, to be developed in the present chapter, are the finite difference methods , obtained by replacing the derivatives in the equation by appropriate numerical differentiation formulae. We thus start with a brief discussion of some elementary finite difference formulae used to numerically approximate first and second order derivatives of functions. We then establish and analyze some of the most basic finite difference schemes for the heat equation, first order transport equations, the second order wave equation, and the Laplace and Poisson equations. As we will learn, not all finite difference approximations lead to accurate numerical schemes, and the issues of stability and convergence must be dealt with in order to distinguish reliable from worthless methods. In fact, inspired by Fourier analysis, the crucial stability criterion follows from how the numerical scheme handles basic complex exponentials. The second category of numerical solution techniques are the finite element methods , which will be the topic of Chapter 11. These two chapters should be regarded as but a pre- liminary foray into this vast and active area of contemporary research. More sophisticated variations and extensions, as well as other types of numerical schemes (spectral, multiscale, Monte Carlo, geometric, symplectic, etc., etc.) can be found in more specialized numerical analysis texts, e.g., [ 73 , 64 , 98 ]. 5.1. Finite Differences. In general, a finite difference approximate to the value of some derivative of a function u ( x ) at a point x in its domain, say u ( x ) or u ( x ), relies on a suitable combination of sampled function values at nearby points. The underlying formalism used to construct these approximation formulae is known as the calculus of finite differences . Its development has a long and influential history, dating back to Newton....

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