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The efficiency of Singly-implicit Runge-Kutta methods for stiff differential equations

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Abstract

Singly-implicit Runge-Kutta methods are considered to be good candidates for stiff problems because of their good stability and high accuracy. The existing methods, SIRK (Singly-implicit Runge-Kutta), DESI (Diagonally Extendable Singly-implicit Runge-Kutta), ESIRK (Effective order Singly-implicit Rung-Kutta) and DESIRE (Diagonally Extended Singly-implicit Runge-Kutta Effective order) methods have been shown to be efficient for stiff differential equations, especially for high dimensional stiff problems. In this paper, we measure the efficiency for the family of singly-implicit Runge-Kutta methods using the local truncation error produced within one single step and the count of number of operations. Verification of the error and the computational costs for these methods using variable stepsize scheme are presented. We show how the numerical results are effected by the designed factors: additional diagonal-implicit stages and effective order.

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  1. Department of Information Technology, Ling Tung University, Taichung, Taiwan

    D. J. L. Chen

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  1. D. J. L. Chen
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Correspondence to D. J. L. Chen.

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The author is also an adjunct professor at the Department of Finance and Risk Management, Ling Tung University and the author’s work is partly supported by the National Science Council of Taiwan

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Chen, D.J.L. The efficiency of Singly-implicit Runge-Kutta methods for stiff differential equations. Numer Algor 65, 533–554 (2014). https://doi.org/10.1007/s11075-013-9797-5

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