In this chapter, we discuss the time-discretization of time-dependent equations. Although the methods apply to general nonlinear time-dependent equations, their analysis is developed in the linear case and, more especially, for the advection-diffusion equation. First, we address the stability of the spectral approximation, namely, the existence of a bounded solution of the differential equations in time resulting from the spectral approximation. Then the major part of the chapter is devoted to the presentation and discussion of the accuracy and stability of the time-discretization schemes. Second- and higher-order methods are considered in the following two cases: two-step methods (essentially based on Backward-Differentiation and Adams-Bashforth schemes) and one-step methods (Runge-Kutta schemes). The chapter ends with a comparison between the different kinds of time-discretization schemes.
KeywordsExplicit Scheme Chebyshev Approximation Chebyshev Collocation Method Fourier Approximation Multistep Scheme
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