Adaptive Approximation of Solutions to Problems with Multiple Layers by Chebyshev Pseudo-Spectral Methods

Abstract

We develop end analyze a family of mappings which enhance the accuracy of Chebyshev pseudo-spectral methods in approximating functions with multiple regions of localized rapid variation (layers). The mapping family depends on 3N - 1 free parameters, where N is the number of layers. N parameters depend upon the locations of the layers and on the widths of the layers, while the other N - 1 parameters depend on the resolution of each layer relative to the first layer. The parameters can be determined adaptively by minimizing a functional which measures the error of the approximation. Techniques to simplify the minimization process are developed. We further demonstrate that the appropriate choice of mappings can lead to a significant reduction in the condition number of matrices associated with Chebyshev pseudo-spectral differentation. We illustrate the effectiveness of the proposed mapping and adaptive procedure by examples in which we approximate (i) given functions exhibiting multiple layers and (ii) the solution of a system of partial differential equations modeling combustion in counterflowing jets so that two distinct flames occur.