A new map for the Chebyshev pseudospectral solution of differential equations with large gradients

Abstract

For problems whose solution has singularities close to [?1,1] in the complex plane, an improvement on the exponential convergence rate of rational spectral collocation methods through enlargement of the ellipse of analyticity of the underlying solution is presented. For time dependent problems, the new clustered Chebyshev points with larger minimal spacing alleviate the time-stepping restriction. In addition, the proposed method can easily be generalized to the approximation of functions with several steep fronts. Numerical results include comparisons of the new method to existing methods, showing the advantage of this new method in comparison with the standard Chebyshev spectral collocation method.