Yonglei Fang

Extended RKN-type methods for numerical integration of perturbed oscillators☆

Abstract In this paper, extended Runge–Kutta–Nystrom-type methods for the numerical integration of perturbed oscillators with low frequencies are presented, which inherit the framework of RKN methods and make full use of the special feature of the true flows for both the internal stages and the updates. Following the approach of J. Butcher, E. Hairer and G. Wanner, we develop a new kind of tree set to derive order conditions for the extended Runge–Kutta–Nystrom-type methods. The numerical stability and phase properties of the new methods are analyzed. Numerical experiments are accompanied to show the applicability and efficiency of our new methods in comparison with some well-known high quality methods proposed in the scientific literature.

Trigonometrically fitted explicit Numerov-type method for periodic IVPs with two frequencies

Abstract In this paper, new trigonometrically fitted Numerov type methods for the periodic initial problems are proposed. These methods are based on the original Numerov-type sixth order method with fifth internal stages motivated by Tsitouras (see [Ch. Tsitouras, Explicit Numerov type methods with reduced number of stages, Comput. Math. Appl. 45 (2003) 37–42]). Some parameters are added to these methods so that they can integrate exactly the combination of trigonometrically functions with two frequencies. Numerical stability and phase properties of the new methods are analyzed. Numerical experiments are carried out to show the efficiency and robustness of our new methods in comparison with the well known codes proposed in the scientific literature.

Order conditions for RKN methods solving general second-order oscillatory systems

This paper proposes and investigates the multidimensional extended Runge-Kutta-Nyström (ERKN) methods for the general second-order oscillatory system y″ + My = f(y, y′) where M is a positive semi-definite matrix containing implicitly the frequencies of the problem. The work forms a natural generalization of our previous work on ERKN methods for the special system y″ + My = f(y) (H. Yang et al. Extended RKN-type methods for numerical integration of perturbed oscillators, Comput. Phys. Comm. 180 (2009) 1777–1794 and X. Wu et al., ERKN integrators for systems of oscillatory second-order differential equations, Comput. Phys. Comm. 181 (2010) 1873–1887). The new ERKN methods, with coefficients depending on the frequency matrix M, incorporate the special structure of the equation brought by the term My into both internal stages and updates. In order to derive the order conditions for the ERKN methods, an extended Nyström tree (EN-tree) theory is established. The results of numerical experiments show that the new ERKN methods are more efficient than the general-purpose RK methods and the adapted RKN methods with the same algebraic order in the literature.

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

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(iωx),exp(−iωx)}

EXPONENTIALLY FITTED TWO-DERIVATIVE RUNGE–KUTTA METHODS FOR THE SCHRÖDINGER EQUATION

\{\exp ({\rm i}\omega x),\exp (-{\rm i}\omega x)\}

New optimized explicit modified RKN methods for the numerical solution of the Schrödinger equation

or equivalently the set {cos(ωx),sin(ωx)}

Extended RKN-type methods with minimal dispersion error for perturbed oscillators

\{\cos (\omega x),\sin (\omega x)\}

A new phase-fitted modified Runge-Kutta pair for the numerical solution of the radial Schrödinger equation

with ω>0

Extended RKN methods with FSAL property for oscillatory systems

\omega >0

A New Optimized Runge-Kutta Pair for the Numerical Solution of the Radial Schrödinger Equation

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.