Numerical Approximation of Model Partial Differential Equations

Abstract

This first chapter is intended as a quick introduction to basic discretization techniques of time-dependent partial differential equations (PDEs). We consider it important that the reader, before tackling the complex problems of the next chapters, have some understanding of the mathematical and physical properties of the following model PDEs: the convection equation, the wave equation, and the heat equation. This chapter is therefore organized as a collection of several short exercises in which model PDEs are theoretically analyzed and numerically solved using the simplest discretization methods. The essential features of numerical methods are presented, with emphasis on fundamental ideas of accuracy, stability, convergence, and numerical dissipation. Particular care is devoted to the validation of numerical procedures by comparing to exact solutions available for these simple cases.