Chapter 9 - Time Integration Methods for Space-discretized Equations

Abstract

This chapter elaborates time integration methods for space-discretized equations. The general formulation of a numerical scheme where time is separated from the space discretization is presented in the chapter. A general methodology for the analysis of the association of a given space discretization with a time integration method can be defined by the steps such as expressing the space discretization into a matrix form including the boundary conditions. It is suggested to select a time integration method for the semi-discretized system of ordinary differential equations and define the methodology for its stability analysis, as a function of the eigenvalues of the space discretization. The stability condition of the time integration scheme has to be compatible with the range of the eigenvalue spectrum—namely, the stability region of the time discretization method must include the whole spectrum of eigenvalues. The matrix representation of the diffusion space operator is analyzed. The eigenvalue spectrum of space-discretized systems is also described in the chapter.