Introduction to Numerical Methods for Time Dependent Differential Equations

CHAPTER 10

LINEAR INITIAL BOUNDARY VALUE PROBLEMS

The purpose of this chapter is to introduce the basic ideas of initial boundary value problems for partial differential equations and their finite difference approximations. We study the model equations of Chapter 9. Energy estimates, the simplest technique to use to study well posedness for these problems, is discussed first. Then the method of lines and the energy estimates for typical semidiscrete approximations are discussed.

10.1 Well-Posed Initial Boundary Value Problems

Fourier mode analysis is not easily generalizable to problems with boundary conditions other than periodic, whereas the construction of energy estimates presents no problem. Energy estimates are thus the classical way to study well posedness of initial boundary value problems.

10.1.1 Heat equation on a strip

Here we study the initial value problem for the heat equation on a strip:

(10.1)

with either fixed Dirichlet boundary conditions, u(0, t) = U₀, u(1, t) = U₁, or homogeneous Neuman boundary conditions, ∂u(0, t)/∂x = ∂u(1, t)/∂x = 0. In the first case it is convenient to redefine the solution u(x, t) by subtracting the linear solution U₀ + x(U₁ − U₀) from it. Then the new function satisfies the same equation but with homogeneous boundary conditions. Therefore, it is sufficient to consider problem (10.1) with either

(10.2)

or

(10.3)

Smoothness of the solution. ...