Trigonometrically fitted two-derivative Runge-Kutta methods for solving oscillatory differential equations

Abstract

Explicit trigonometrically fitted two-derivative Runge-Kutta (TFTDRK) methods solving second-order differential equations with oscillatory solutions are constructed. When the second derivative is available, TDRK methods can attain one algebraic order higher than Runge-Kutta methods of the same number of stages. TFTDRK methods have the favorable feature that they integrate exactly first-order systems whose solutions are linear combinations of functions from the set { exp ( i ? x ) , exp ( ? i ? x ) } $\{\exp ({\rm i}\omega x),\exp (-{\rm i}\omega x)\}$ or equivalently the set { cos ( ? x ) , sin ( ? x ) } $\{\cos (\omega x),\sin (\omega x)\}$ with ? > 0 $\omega >0$ the principal frequency of the problem. Four practical TFTDRK methods are constructed. Numerical stability and phase properties of the new methods are examined. Numerical results are reported to show the robustness and competence of the new methods compared with some highly efficient methods in the recent literature.