10: Numerical Solutions of PDEs

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There’s no sense in being precise when you don’t even know what you’re talking about. - John von Neumann (1903-1957)

Most of the book has dealt with finding exact solutions to some generic problems. However, most problems of interest cannot be solved exactly. The heat, wave, and Laplace equations are linear partial differential equations and can be solved using separation of variables in geometries in which the Laplacian is separable. However, once we introduce nonlinearities, or complicated non-constant coefficients intro the equations, some of these methods do not work. Even when separation of variables or the method of eigenfunction expansions gave us exact results, the computation of the resulting series had to be done on a computer and inevitably one could only use a finite number of terms of the expansion. So, therefore, it is sometimes useful to be able to solve differential equations numerically.

In this chapter we will introduce the idea of numerical solutions of partial differential equations. However, we will first begin with a discussion of the solution of ordinary differential equations in order to get a feel for some common problems in the solution of differential equations and the notion of convergence rates of numerical schemes. Then, we turn to the finite difference method and the ideas of stability. Other common approaches may be added later.

10.1: Ordinary Differential Equations
In this section we will look at the simplest method for solving first order equations, Euler’s Method. While it is not the most efficient method, it does provide us with a picture of how one proceeds and can be improved by introducing better techniques, which are typically covered in a numerical analysis text.
10.2: The Heat Equation
THE HEAT EQUATION CAN BE SOLVED USING SEPARATION OF VARIABLES. However, many partial differential equations cannot be solved exactly and one needs to turn to numerical solutions. The heat equation is a simple test case for using numerical methods. Here we will use the simplest method, finite differences.
10.3: Truncation Error
10.4: Stability