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Author Verner, J. H.
Source CiteSeerX
Content type Text
File Format PDF
Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Subject Keyword Two-step Runge Kutta Method ♦ Order Condition ♦ Tsrk Method ♦ Starting Method ♦ Am Subject Classications ♦ Runge Kutta Method ♦ Tsrk Formula ♦ Two-step Runge-kutta ♦ Correct Value ♦ Several Nonlinear Test Problem ♦ Key Word ♦ Perturbed Order Condition ♦ Local Error Estimation ♦ Rst Step ♦ Error Estimator ♦ Exact Local Truncation Error ♦ Intricate Polynomial Equation ♦ Complete Set ♦ Complementary Set
Abstract In [5], Jackiewicz and Verner derived formulas for, and tested the im-plementation of two-step Runge-Kutta (TSRK) pairs. For pairs of orders 3 and 4, the error estimator accurately tracked the exact local truncation error on several nonlinear test problems. However, for pairs designed to achieve order 8, the results appeared to be only of order 6. This deciency was identied in [2] by Hairer and Wanner who used B-series to formulate a complete set of order conditions for TSRK methods, and showed that if the order of a TSRK method is at least two greater than its stage-order, special starting values are necessary for the rst step. In [8], Verner showed that such starting values have to be perturbed from their asymptotically correct values to include errors of precisely the form which the selected TSRK formula is designed to propagate from step to step. It was shown that starting values could be obtained by solving the problem using a complementary set of Runge{Kutta methods which satised perturbed order conditions to obtain perturbed starting values, and that such methods could be obtained by solving these order conditions directly. The design used there required solving an intricate polynomial equation. Here, the design is improved, and new starting methods are simpler to derive, and perhaps may lead to starting methods for TSRK methods of order 8. Key words. two-step Runge{Kutta methods, starting methods, order conditions, local error estimation, implementation. AMS subject classications. 65L05, 65L06, 65L20.
Educational Role Student ♦ Teacher
Age Range above 22 year
Educational Use Research
Education Level UG and PG ♦ Career/Technical Study
Publisher Date 2004-01-01