Well-posedness and Stability

Abstract

Stability is a fundamental concept for any type of PDE approximation. A stable approximation is such that small perturbations in the given data cause only small perturbations in the solutions. Furthermore, the solutions converge to the true solution of the PDE as the step size h tends to zero. The extra condition required for this property is that the PDE problem is well posed. In this chapter we shall present a survey of the basic theory for the well-posedness and stability. The theory can be divided into three different techniques: Fourier analysis for Cauchy and periodic problems, the energy method and Laplace analysis (also called normal mode analysis) for initial-boundary value problems. In order to emphasize the similarities between the continuous and discrete case, we treat the application of each technique to both the PDE and the finite difference approximations in the same section (the Laplace technique for PDE is omitted). 2.1 Well Posed Problems We consider a general initial-boundary value problem ∂ u ∂t = Pu + F 0 t Bu = g u = f t = 0 (2.1) Here P is a differential operator in space, and B is a boundary operator acting on the solution at the spacial boundary. (Throughout this book, we will refer to t as the time coordinate, and to the remaining independent variables as the space variables, even if the physical meaning may be different.) There are three types of data that are fed into the problem: F is a given forcing function , g is a boundary function and f is an initial function. (By "function" we mean here the more general concept "vector function", i.e., we are considering systems of PDE.) A well posed problem 13