Temporal Discretization

Abstract

In most applications of spectral methods to partial differential equations the spatial discretization is spectral but the temporal discretization uses conventional finite-differences. The typical evolution equation can be written

$$ \begin{gathered} \frac{{\partial u}}{{\partial t}} = f\left( {u,t} \right)t > 0 \hfill \\ u\left( 0 \right) = 0, \hfill \\ \end{gathered} $$

(4.1.1)

where the (generally) non-linear operator f contains the spatial part of the PDE. (The dependence of f upon t will not be indicated explicitly hereafter except when it is essential.) Following the general formulation of Chap. 3 the semi-discrete version is

$$ {Q_N}\frac{{d{u^N}}}{{dt}} = {Q_N}{f_N}\left( {{u^N}} \right), $$

where u^N is the spectral approximation to u,f _N denotes the spectral approximation to the operator f,and Q _N is the projection operator which characterizes the scheme. Let us set U(t) = Q _N u^N (t). For example, in a collocation method for a Dirichlet boundary value problem, U(t) represents the set of the interior grid values of u^N (t). Then, the previous discrete problem can be written in the form

$$ \frac{{dU}}{{dt}} = F\left( U \right). $$

(4.1.2)