Spectral Methods in Time for Parabolic Problems

Abstract

A pseudospectral explicit scheme for solving linear, periodic, parabolic problems is described. It has infinite accuracy both in time and in space. The high accuracy is achieved while the time resolution parameter $M(M = O({1 / {\Delta t}})$ for time marching algorithm) and the space resolution parameter $N(N = O({1 / {\Delta x))}}$ must satisfy $M = O(N^{1 + \varepsilon } )\varepsilon > 0$, compared to the common stability condition $M = O(N^2 )$, which must be satisfied in any explicit finite-order time algorithm.