ABSTRACT
The majority of practical problems involving partial differential equation (PDE) cannot be solved by analytic methods. Therefore it is expedient to study numerical methods, which lead to approximate solutions of PDEs. In many introductory texts, the student is confronted with several approximation or iteration schemes, and then the student is simply asked to write computer programs to implement these schemes. This chapter introduces some basic difference equations to study the propagation of round-off errors and to handle the problem of determining mesh size which yields the greatest accuracy, in the presence of round-off errors. It discusses the explicit difference method, using heat equation as a model. The chapter provides an overview of some other numerical methods for PDEs. It examines the nature of the discretization error and proves a convergence theorem for the explicit difference method. The chapter considers the method of lines for problems in higher dimensions, and the Rayleigh—Ritz approximation for solutions of certain linear boundary value problems.
