A Modified Runge-Kutta Method for Nonlinear Dynamical Systems with Conserved Quantities

[+] Author and Article Information
Guang-Da Hu

Professor, Department of Mathematics Shanghai University, Shanghai, 200444, China

1Corresponding author.

ASME doi:10.1115/1.4036761 History: Received November 28, 2016; Revised May 10, 2017


In this paper, explicit Runge-Kutta methods are investigated for numerical solutions of nonlinear dynamical systems with conserved quantities. The concept, $\varepsilon-$preserving is introduced to describe the conserved quantities being approximately retained. Then a modified version of explicit Runge-Kutta methods based on the optimization technique is presented. With respect to the computational effort, the modified Runge-Kutta method is superior to implicit numerical methods in literature. The order of the modified Runge-Kutta method is the same as the standard Runge-Kutta method, but it is superior in preserving the conserved quantities to the standard one. Numerical experiments are provided to illustrate the effectiveness of the modified Runge-Kutta method.

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