Using Green's functions and split Gaussian integration for two-point boundary value problems on the ray

Abstract

We continue our study of the construction of numerical methods for solving two-point boundary value problems using Green's functions, building on the successful use of product integration to achieve the convergence expected of Gauss-type quadrature schemes. We introduce refinements such as the use of cardinal basis functions to eliminate the need for a transformation from the ordinates to the expansion coefficients. For problems on the ray, we algebraically map the ray to a segment, and there use (cardinal) Legendre polynomials for interpolation and Gauss's rule for quadrature. Numerical examples (the heat and Burgers equations) demonstrate the applicability of the method to problems on the ray, particularly for the sequence of two-point boundary value problems arising from constant time-stepping for parabolic evolution problems; nonlinear terms, as in the Burgers equation, present no special difficulty.