VSVO formulation of the Taylor method for the numerical solution of ODEs. (English) Zbl 1085.65056
Summary: This paper presents an analysis of the Taylor method for the numerical solution of ordinary differential equations (ODEs) when a very high precision of the solution is required. Some theoretical properties of the Taylor method are considered. From the practical point of view a variable-stepsize variable-order (VSVO) scheme is presented and its utility is discussed with several examples. To reach the goal of high precision the use of multiprecision libraries is considered. Finally, some numerical tests based on the test problems given by W. H. Enright and J. D. Pryce [ACM Trans. Math. Softw. 13, 1–27 (1987; Zbl 0617.65069)] and on a set of important problems in dynamical systems and astrodynamics are presented showing the benefits of the VSVO formulation, especially for high-precision demands, compared with a well established Runge-Kutta code.
MSC:
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 34A34 | Nonlinear ordinary differential equations and systems |
| 65L50 | Mesh generation, refinement, and adaptive methods for ordinary differential equations |
Citations:
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