The use of symbolic manipulation to enable the practical use of power series integrators

Abstract

A prototype scientific programming environment for solving initial and boundary value problems in ordinary differential and integro-differential equations is presented. This environment is designed to assist scientific programmers in overcoming the semantic gap between the formulation of the mathematical model of a physical system and the code for solving that model. The environment is part of a larger view of literate programming than has been proposed and is designed to interact with WEB systems. The inputs to the environment are a standardized description of the mathematical system of differential, integro-differential, and algebraic equations. It is desired to use a power series integrator. This integrator is: (1) faster than standard Runge-Kutta or multistep integrators, (2) robust in that it works equally well with stiff and non-stiff problems, and (3) convenient and fast in solving problems with convolution integral terms. A parser has been created to convert the standardized description into statements in a typical high level language for inclusion into a computer program. The use of these integrators has previously required significant manual algebra. The prototype system developed and reported here removes this manual requirement. Conclusions and recommendations for further study and how to use the lessons learned in this narrow study on the more general scientific programming issues are included. This research is typical of the radically different "reconsiderations" which may lead to better methods for the newer architectures that are becoming available for scientific computing.