Applied Mathematics for Science and Engineering

8Numerical Solution of Partial Differential Equations

Introduction

Our usual approach will involve discretization of partial differential equations (PDEs), followed by solution of the resulting algebraic equations. Discretization is key to both finite-difference methods (FDMs) and finite-element methods (FEMs). The two approaches require the same level of numerical effort, but the latter is particularly useful for problems involving irregular shapes and boundaries (an introduction to FEM will be provided in Chapter 11). On the other hand, FDMs are much less software-dependent, and for simple problems, FDM solutions can be obtained with a broad spectrum of hardware–software combinations, even through the use of commonplace tools like spreadsheet programs. Thus, the analyst can solve many important practical problems without commercial modeling software, without high-level language proficiency, without compiler experience, and without mesh generation and refinement.

We should anticipate that when we solve a PDE numerically, we may not obtain a completely accurate solution. Of course, we expect discrepancies arising from both roundoff and truncation, and a common view is that we are solving the given PDE with some acceptable level of error. There is a second viewpoint that is useful in the context of certain computations, and it reveals a more insidious problem that we need to recognize: When we discretize a PDE, we are actually creating a PDE that may have additional terms; that is, ...