Part 2 Lectures In basic computational numerical analysis has contents: Numerical solution of ODEs, numerical solution of PDEs (mathematical introduction, overview of discretization methods for PDEs, elliptic equations,.). | Chapter 4 Numerical Solution of Ordinary Differential Equations Development of the ability to solve essentially any ordinary differential equation (ODE) on the digital computer is one of the major achievements of modern numerical analysis. While it is not the case that absolutely any and every ODE problem can be solved easily, it is true that the great majority of problems that occur in practice can be solved quite efficiently with a well-chosen method. This is true generally, independent of whether the problems are linear or nonlinear. There are two types of problems associated with ODEs: i) initial-value problems (IVPs), and ii) boundary-value problems (BVPs). Numerical methods for IVPs are highly developed, and except for extremely large systems or problems involving very high-frequency oscillations, there exist many standard software packages that can be readily used. The main purpose of discussions here will be to treat very basic methods that will provide insight into construction of such software. The situation is not quite so favorable for BVPs, especially for nonlinear problems and for systems. One of the main approaches is to convert these problems to IVPs, and then use the initial-value packages. This, however, has no guarantee of success; our approach to BVPs will be to use techniques developed specifically for such problems. These have been demonstrated to be very effective, and with the aid of Richardson extrapolation can be highly accurate. Moreover, they have direct application to construction of methods for numerical solution of partial differential equation BVPs. The chapter is subdivided into two main sections, each providing treatment of one of the main types of ODE problems: Sec. 1 will deal with initial-value problems, and Sec. 2 with boundary-value problems. Initial-Value Problems In this section we discuss some of the main methods for solving initial-value problems for ordinary differential equations. We begin with a brief mathematical .
