Solving Ordinary Differential Equations Using Taylor Series

Abstract

Taylor series methods compute a solution to an initial value problem in ordinary differential equations by expanding each component of the solution in a long series. A portable translator program accepts statements of the system of differential equations and produces a portable FORTRAN object code which is then run to solve the system. At each step of the integration, the object program generates the series for each component of the solution, analyzes that series to determine the optimal step, and extends the solution by analytic continuation. The translator is easy to use, yet it is powerful and flexible. The computer time required by this approach consists of time to run the translator plus time to run the object code, CPU time and storage requirements depend on the size and complexity of the system of ODEs. Theoretical estimates and empirical test results are given for Hull's test problems, and comparisons with DVERK and DGEAR from IMSL are given. The computer time for all preproeessmg, compilation, and linking Is included Taylor series method executes faster and yields a more accurate answer than the standard methods for most of the problems in the test set. The Taylor series method is most attractwe for small systems and for stringent accuracy tolerances. © 1982, ACM. All rights reserved.