Two-derivative Runge-Kutta-Nyström methods for second-order ordinary differential equations

Classical Runge-Kutta-Nyström (RKN) methods for second-order ordinary differential equations are extended to two-derivative Runge-Kutta-Nyström (TDRKN) methods involving the third derivative of the solution. A new version of Nyström tree theory and the corresponding B-series theory are developed, based on which the order conditions for TDRKN methods are derived. A two-stage explicit TDRKN method of order four and a three-stage explicit TDRKN method of order five are constructed. The linear stability of the new methods is analyzed. The results of numerical experiments show that the new TDRKN methods are more efficient than the traditional RKN methods of the same algebraic order.

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