Approximation Methods for Initial Value Problems in Partial Differential Equations

Abstract

We begin this rather extensive chapter by presenting several numerical methods for solving the heat equation and the wave equation (Section 4.1 to 4.3), which are typical examples of parabolic and hyperbolic problems, respectively. The methods we discuss comprise not only finite-difference methods but also Galerkin methods; our methods are either explicit or implicit and include so-called multilevel (more precisely, three-level) methods. In Section 4.4, we present finite-difference and Galerkin methods for approximating various classes of nonlinear initial value problems and discuss the solvability of the associated systems of nonlinear equations. Finally, we show in Section 4.5 how the problems considered in the previous sections — along with their approximating equations — can be viewed as operator equations in appropriate function spaces.