Solution Methods for Parabolic Partial Differential Equations

Abstract

Introduction Initial developments in computing were dominated by two classes of problems: (i) the jury or the boundary value problems – typically classified as elliptic partial differential equation in Chapter 3; (ii) the evolution or the initial–boundary value problems which are represented by parabolic and hyperbolic partial differential equations. In fact, the solution methods for heat equation (a parabolic partial differential equation) were central to the early development of the subject. These classical approaches are discussed in this chapter, with additional insight brought through spectral analysis of the schemes. It is noted that the stability analysis of numerical schemes was developed with respect to heat equation by von Neumann, as described in [41, 53]. This was considered a major milestone in the development of the subject. But, the readers' attention is also drawn to the correct analysis advanced recently, as described in [259] and Chapter 8, with respect to 1D convection equation. In fluid dynamics, a major milestone was the introduction of boundary layer concept by Ludwig Prandtl in 1904, which dominated fluid dynamics studies. Readers are referred to [209] for details of the development. Boundary layer equation is an example of parabolic partial differential equation.