# Optimal use of a numerical method for solving differtial equations based on Taylor-series expansions

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## Abstract

Optimal use of a numerical method for solving differential equations based on Taylor series expansions INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 32,471-499 (1991) OPTIMAL USE OF A NUMERICAL METHOD FOR SOLVING DIFFERENTIAL EQUATIONS BASED ON TAYLOR SERIES EXPANSIONS P. J. M. SONNEMANS, L. P. H. DE GOEY AND J. K. NIEUWENHUIZEN Eindhoven University of Technology, Faculty of Mechanical Engineering, P.O. Box 513,5600 M B Eindhoven, The Netherlands SUMMARY Efficiency in solving differential equations is improved by increasing the order of a Taylor series approxima- tion. Computing time can be reduced up to a factor of 40 and an amount of memory storage can be saved, up to a factor of 70. The truncation error can be estimated not only by order but also by magnitude. 1. INTRODUCTION Solving problems numerically in engineering today often requires huge amounts of computing time and memory storage: for example, three-dimensional problems involving non-stationary phenomena and complex geometry. Therefore it is of the utmost importance to develop numer- ical methods which are as efficient as possible for the problem to be tackled. A method for solving differential equations is proposed. The method, based on the continuity of the solution, is not new. '- Gibbons' formulated automatic programs to integrate ordinary differential equations which are reducible to rational form. Barton et al. 2,3 showed the efficiency of the method in terms of required computing time, concluding that in some applications computing time may be reduced up to a factor of 40. For heat conduction the method is superior to all others in shortness and accuracy, according to C01latz.~ He generalized the application to two-dimensional systems and pointed out that instability was one of the problems still to be tackled. In a stability analysis, Small5-* indicated that the method is Von Neumann stable over a wide class of equations, if the differential equations are line

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