SIAM Journal on Numerical Analysis
High-order P-stable multistep methods
Journal of Computational and Applied Mathematics
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Exponentially fitted Runge-Kutta methods
Journal of Computational and Applied Mathematics - Special issue on numerical anaylsis 2000 Vol. VI: Ordinary differential equations and integral equations
Exponentially fitted explicit Runge-Kutta-Nyström methods
Journal of Computational and Applied Mathematics
Runge-Kutta methods adapted to the numerical integration of oscillatory problems
Applied Numerical Mathematics
A class of explicit two-step hybrid methods for second-order IVPs
Journal of Computational and Applied Mathematics
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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 驴 x ) , exp ( 驴 i 驴 x ) } $\{\exp ({\rm i}\omega x),\exp (-{\rm i}\omega x)\}$ or equivalently the set { cos ( 驴 x ) , sin ( 驴 x ) } $\{\cos (\omega x),\sin (\omega x)\}$ with 驴 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.