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

Authors:
Truong Nguyen-Ba;Vladan Bozic;Emmanuel Kengne;Rémi Vaillancourt
Affiliations:
Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario, Canada K1N 6N5;Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario, Canada K1N 6N5;Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario, Canada K1N 6N5;Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario, Canada K1N 6N5
Venue:
Journal of Computational and Applied Mathematics
Year:
2010

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Abstract

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.