Xiong You

A New Trigonometrically Fitted Two-Derivative Runge-Kutta Method for the Numerical Solution of the Schrödinger Equation and Related Problems

A new trigonometrically fitted fifth-order two-derivative Runge-Kutta method with nvariable nodes is developed for the numerical solution of the radial Schrodinger equation nand related oscillatory problems. Linear stability and phase properties of the new nmethod are examined. Numerical results are reported to show the robustness and competence nof the new method compared with some highly efficient methods in the recent nliterature.

Novel phase-fitted symmetric splitting methods for chemical oscillators

Splitting strategy is considered for the numerical solution of oscillatory chemical reaction systems. When applied to the harmonic oscillator, traditional splitting methods with constant coefficients are shown to be have some order of phase lag though they are zero-dissipative. Phase-fitted symmetric splitting methods of order two and order four are constructed. The result of the numerical experiment on the Lotka–Volterra system shows that the new phase-fitted symmetric splitting methods are more effective than their prototype splitting methods and can preserve the invariant of the system in long-term compared with the classical Runge–Kutta method.

Adiabatic Filon-type methods for highly oscillatory second-order ordinary differential equations

In this paper, we study efficient numerical integrators for linear and nonlinear systems of highly oscillatory second-order ordinary differential equations. The systems are reformulated as a first-order system, which is then transformed to adiabatic variables. The solution of the transformed system is a smoother function which is more accessible to numerical approximation than the original system. We develop Filon-type methods for linear systems by approximating the integral as a linear combination of function values and derivatives. We then present a special combination of Filon-type methods and waveform relaxation methods for nonlinear systems. Both types of methods can be used with far larger step sizes than those required by traditional schemes and their performance drastically improves as frequency grows, as are illustrated by numerical experiments.

New optimized two-derivative Runge-Kutta type methods for solving the radial Schrödinger equation

Abstract This paper focuses on adapted two-derivative Runge-Kutta (TDRK) type methods for solving the Schrödinger equation. Two new TDRK methods are derived by nullifying their phase-lags and the first derivatives of the phase-lags. Error analysis is carried out by means of asymptotic expressions of the local errors. Numerical results are reported to show the efficiency and robustness of the new methods in comparison with some RK type methods specially tuned to the integration of the radial time-independent Schrödinger equation with the Woods–Saxon potential.

On modified TDRKN methods for second-order systems of differential equations

ABSTRACT This paper concerns modified Two-Derivative Runge–Kutta–Nyström (TDRKN) methods for solving second-order initial value problems. Compromised with the deduced order and symmetry criteria, two implicit and forth-order two-stage TDRKN schemes are derived through a mixed collocation approach. Phase and periodic stability features are examined. Numerical experiments are carried out to illustrate the effectiveness and competence of our new methods. Comparisons with existing highly accurate and efficient numerical methods are given.

Exponentially fitted symmetric and symplectic DIRK methods for oscillatory Hamiltonian systems

The order conditions for modified Runge–Kutta methods are derived via the rooted trees. Symmetry and symplecticity conditions and exponential fitting conditions for modified diagonally implicit Runge–Kutta (DIRK) are considered. Three new exponentially fitted symmetric and symplectic diagonally implicit Runge–Kutta (EFSSDIRK) methods of respective second order and fourth order are constructed. Phase properties of the new methods are analyzed. The new EFSSDIRK methods are applied to several Hamiltonian problems and compared to the results obtained by the existing symplectic DIRK methods in the literature.

An adapted explicit hybrid four-step method for the numerical solution of perturbed oscillators

In this paper, the construction of an adapted explicit hybrid four-step method for the numerical integration of perturbed oscillators is investigated. This four-step method is based on the algorithm of Scheifele which is obtained by refining the classical method of power series. The local truncation error, phase properties and linear stability of the new method are analyzed. Numerical experiments are reported to show the high accuracy and efficiency of the new method when it is compared with some high-quality methods recently proposed in the literature.

Modified two-derivative Runge–Kutta methods for the Schrödinger equation

A family of modified two-derivative Runge–Kutta (MTDRK) methods for the integration of the Schrödinger equation are obtained. Two new three-stage and fifth order TDRK methods are derived. The numerical results in the integration of the radial Schrödinger equation with the Woods–Saxon potential are reported to show the high efficiency of our new methods. The results of the error analysis are illustrated by the resonance problem.

An explicit trigonometrically fitted Runge–Kutta method for stiff and oscillatory problems with two frequencies

ABSTRACT In this paper, a new explicit fourth-order Runge–Kutta method adapted to oscillatory problems with two frequencies (μ and ν) is developed. The new method can integrate exactly the different harmonic oscillators and simultaneously. Numerical stability and phase properties of the new method are analysed. Numerical experiments are conducted to illustrate the high accuracy and competence of our new method in comparison with some highly effective methods from the recent literature.

Runge–Kutta–Nyström methods with equation dependent coefficients and reduced phase lag for oscillatory problems

A new family of one-parameter equation dependent Runge–Kutta–Nyström (EDRKN) methods for the numerical solution of second–order differential equations are investigated. The coefficients of new three-stage EDRKN methods are obtained by nullifying up to appropriate order of moments of operators related to the internal and external stages. A fifth-order EDRKN method that is dispersive of order six and dissipative of order five and a fourth-order EDRKN method that is dispersive of order four and zero-dissipative are derived. Phase analysis shows that there exist no explicit EDRKN methods that are P-stable. Numerical experiments are reported to show the high accuracy and efficiency of the new EDRKN methods.