A one-step 7-stage Hermite-Birkhoff-Taylor ODE solver of order 11

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:234

Article

Nguyen-Ba, Truong¹  Bozic, Vladan¹  Kengne, Emmanuel¹  Vaillancourt, Remi¹ 

[1] Univ Ottawa, Dept Math & Stat, Ottawa, ON K1N 6N5, Canada

关键词: General linear method;    Hermite-Birkhoff method;    Vandermonde-type systems;    Maximum global error;    Number of function evaluations;    CPU time;    DP87;    Comparing ODE solvers;   

DOI  :  10.1016/j.cam.2009.12.015

来源: Elsevier

【 摘 要 】

A one-step 7-stage Hermite-Birkhoff-Taylor method of order 11, denoted by HBT(11)7, is constructed for solving nonstiff first-order initial value problems y' = f (t, y). y(t(0)) = y(0). The method adds the derivatives y' to y((6)), used in Taylor methods, to a 7-stage Runge-Kutta method of order 6. Forcing an expansion of the numerical solution to agree with a Taylor expansion of the true solution to order 11 leads to Taylor- and Runge-Kutta-type order conditions. These conditions are reorganized into Vandermonde-type linear systems whose solutions are the coefficients of the method. The new method has a larger scaled interval of absolute stability than the Dormand-Prince DP87 and a larger unscaled interval of absolute stability than the Taylor method, T11, of order 11. HBT(11)7 is superior to DP87 and T11 in solving several problems often used to test higher-order ODE solvers on the basis of the number of steps, CPU time, and maximum global error. Numerical results show the benefit of adding high-order derivatives to Runge-Kutta methods. (C) 2010 Elsevier B.V. All rights reserved.

【 授权许可】

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