Collocation Methods for Parabolic Partial Differential Equations in One Space Dimension

Abstract

Collocation at Gaussian points for a scalar m'th order ordinary differential equation has heen studied by C. de Boor and B. Swartz, J. Douglas, Jr. and T. Dupont, using collocation at Gaussian points, and a combination of "energy estimates" and approximation theory have given a comprehensive theory for parabolic problems in a single space variable. While the results of this report parallel those of Douglas and Dupont, the approach is basically different. The Laplace transform is used to "lift" the results of de Boor and Swartz to linear parabolic problems. This indicates a general procedure that may be used to "lift" schemes for elliptic problems to schemes for parabolicc problems. Additionally there is a section on longtime integration and A-stability.