A two-dimensional adaptive pseudo-spectral method

We develop a two-dimensional adaptive pseudo-spectral procedure which is capable of improving the approximation of functions which are rapidly varying in two dimensions. The method is based on introducing two-dimensional coordinate transformations chosen to minimize certain functionals of the solution to be approximated. The method is illustrated by numerical computation of the solutions to a system of reaction diffusion equations modeling the gasless combustion of a solid fuel. Spatio-temporal patterns are computed as a parameter μ, related to the activation energy, is increased above a critical value μ_c. The spatial patterns are characterized by a very rapid variation in the direction of the axis of the cylinder, together with a standing wave pattern in the direction of the azimuthal angle ψ For small values of μ - μ_c the solutions exhibit a nearly sinusoidal dependence in both time and ψ As μ is increased further relaxation oscillations in both time and ψ occur. Beyond a critical value of μ stable time-periodic solutions are no longer found and the solution exhibits a quasi-periodic time dependence.