Spectral methods for the solution of nonlinear boundary value problems, a case study

Abstract

Spectral methods enjoy a variety of well known virtues for the solution of ordinary and partial differential equations. The coefficient spectrum falls off exponentially (as for the use of any basis set which is the solution of a singular Sturm-Liouville problem). Nonlinear terms can be effectively accomodated by a true spectral calculation in one dimensional problems, or two dimensional problems with modest resolution, and pseudospectrally by use of the FFT algorithm for larger two dimensional or three dimensional problems. Boundary conditions are frequently easily imposed by natural means of the tau method as an obvious (linear) algebraic constraint equation on the coefficients. Since spectral methods are global in character, and infinite in order, no special techniques are required to calculate derivatives at boundaries (unlike finite difference algorithms). Finally, a variety of diagnostic integral tests of the accuracy of the solution are usually easily implemented. For the model problem considered in this talk, a nonlinear boundary value problem, conventional shooting methods may encounter several difficulties including the occurrence of spontaneous singularities, and intrinsic instability of the desired solution. Numerical solution of this problem using the Gear package, an adaptive step Romberg integration routine of the author, and a spectral program were implemented. (Details will be discussed in a forthcoming article by Ruehr and Ierley, 1985.) For accuracy and efficiency spectral methods proved obviously superior (unless quite high accuracy is required of high derivative terms near the origin). Greater programming time is, for this problem, the only significant drawback.