Department of Mathematics and Statistics

Simon Fraser University

Burnaby, British Columbia, Canada V5A 1S6

Abstract

We discretize first order equations
$$*
u_t + c(x,t) u_x + b(x,t) u = f(x,t)
*
subject to periodic boundary conditions
(in $$*x* and $$*t*)
with a Fourier collocation method.
The matrix of the resulting linear system
is moderately sparse and highly structured.
Iterative methods are an excellent choice for solving
such systems. We compare the
performance of various iterative solvers,
and propose an effective preconditioning strategy
for this class of matrices. The method extends
easily to nonlinear PDEs, where each Newton step
requires the solution of a linear first order
equation. One such problem, namely the computation of
an invariant torus, will be presented as an example.