Pseudospectral Methods and Composite Complex Maps for Near-Boundary Critical Points

Abstract

The rate of convergence of the pseudospectral approximation to singular linear differential eigenproblems is asymptotically geometric, but is often seriously weakened by the presence of singulari_ties, called critical points or critical latitudes. One remedy is to implement an independent variable transformation which distorts the computational domain into the complex plane and away from the critical point. These complex maps can then be chosen to minimize the effect of the critical points. However, the degree of improvement is limited for critical points near a boundary point, since each contour produced by the complex maps must terminate there to enforce the boundary conditions. In this paper, new complex maps are developed for problems containing a single near-boundary critical point. These new composite complex maps are polynomials of degree 2p, wherep≥ 1 is the level of composition. Formulae for the optimal map parameters are deduced analytically and indicate that significant acceleration of the geometric rate of convergence is possible. A test problem is solved to illustrate the technique. Although successful, it is shown that previously ignored algebraic factors in the formula for the error may become significant when utilizing composite complex maps.