Mathematical Modeling with Multidisciplinary Applications

CHAPTER 3

NUMERICAL METHODS: AN INTRODUCTION

XIN-SHE YANG

School of Science and Technology, Middlesex University, London, UK

Because it is not always possible to solve differential equations analytically, numerical methods have become an important tool in modeling and simulation. In fact, computational modeling has become the so-called third paradigm, complementing the tradition theoretical and experimental approaches to problem solving. Among many powerful numerical methods, the finite difference method is one of the most popular methods that are used commonly in computer simulations. It has the advantage of simplicity and clarity, especially in 1D configurations and other cases with regular geometry. The finite difference method essentially transforms an ordinary differential equation into a coupled set of algebraic equations by replacing the continuous derivatives with finite difference approximations on a grid of mesh or node points that span the domain of interest based on the Taylor series expansions. In general, the boundary conditions and boundary nodes need special treatment.

3.1 DIRECT INTEGRATION

The second-order or higher-order ordinary differential equations can be written as a first-order system of ODEs. Since the technique for solving a system is essentially the same as that for solving a single equation

(3.1)

we shall focus on the first-order equation in the rest ...