ON THE ASYMPTOTIC EXPANSIONS OF NUMERICAL SOLUTIONS TO SECOND ORDER DIFFERENTIAL EQUATIONS WITH A REGULAR SINGULAR POINT

Abstract

When difference schemes with uniformly spaced gridpoints are applied to second order ordinary differential equations with a regular singular point, it is often the case that the resulting numerical approximation does not have a uniform asymptotic expansion. As a consequence, postprocessing, such as h2-extrapolation is not an option. This paper examines the cause of this phenomenon and finds that the existence of such expansions requires the discretization of the boundary conditions at the singular point to be compatible with the discretization of the differential equation. In addition, it is shown how an understanding of the need for compatible discretization can assist in the construction of schemes for several classes of equations that arise when symmetry is used to reduce partial differential equations to ordinary differential equations with a regular singular point.