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

We are interested in initial value problems related to systems of first-order ordinary differential equations (ODEs) of the form

Recently, Simos et al. [

Inspired by the approaches of Simos [

We start with a special form of explicit two-derivative Runge-Kutta (TDRK) methods proposed by Chan and Tsai [

According to Chan and Tsai [

Order 2 requires

Order 3 requires, in addition,

Order 4 requires, in addition,

Order 5 requires, in addition,

In order to adapt TDRK methods to the problem (

If we require the TDRK method (

For small values of

It can be seen that when

In this section, we investigate the linear stability of the new method. We consider the following test equation:

For the TDRK method (

The imaginary stability region of the TFTDRKV5 is plotted in Figure

Stability region of method TFTDRKV5.

For the TDRK method (

Denoting the ratio

In this section we carry out six numerical results to illustrate the performance of the new method. The criterion used in the numerical comparison is the decimal logarithm of the maximum global error (LOG10(ERROR)) versus the computational effort measured by the CPU seconds (CPU SECONDS) required by each problem. The methods used for comparison are listed in the following.

EFRK4: the fourth-order exponentially fitted Runge-Kutta method given by Vanden Berghe et al. in [

RK4V: the fourth-order optimized Runge-Kutta method given by Van de Vyver in [

RK5S: the fifth-order trigonometrically fitted Runge-Kutta method derived by Anastassi and Simos in [

RK4S: the fourth-order exponentially fitted Runge-Kutta method derived by Simos in [

EFRK5: the fifth-order trigonometrically fitted Runge-Kutta method derived by Sakas and Simos in [

TFTDRKV5: the fifth-order TFTDRK method with one evaluation of function

Consider the numerical integration of the Schrödinger equation

Efficiency curves for Problem

Efficiency curves for Problem

Efficiency curves for Problem

Efficiency curves for Problem

We consider the inhomogeneous equation

Efficiency curves for Problem

We consider the linear problem studied in [

Efficiency curves for Problem

We consider the following “almost periodic” orbit problem studied by Franco and Palacios [

In this test we take

Efficiency curves for Problem

We consider the two-dimensional problem studied in [

Efficiency curves for Problem

We consider the Fermi-Pasta-Ulam problem studied in [

The numerical results given in Figure

In Figures

Efficiency curves for Problem

A new trigonometrically fitted two-derivative Runge-Kutta method of algebraic order five is derived in this paper. We also analyze the linear stability and phase properties of the new method. Numerical results are reported to show the robustness and competence of the new method compared with some highly efficient methods in the recent literature. To explain the superiority of the new method, we observe that, compared with RK methods of a specific number of stages, TDRK methods have the possibility of gaining a higher algebraic order than traditional RK methods with the same number of stages. We conclude that, for problems with oscillatory solutions, trigonometrically fitted TDRK methods are more accurate and more efficient than the adapted (exponentially or trigonometrically fitted) RK methods.

The authors are grateful to the editors and the anonymous referees for their constructive comments and valuable suggestions which have helped to improve the paper. This research was partially supported by NSFC (no. 11101357 and no. 11171155), the foundation of Shandong Outstanding Young Scientists Award Project (no. BS2010SF031), the foundation of Scientific Research Project of Shandong Universities (no. J13LI03), NSF of Shandong Province, China (no. ZR2011AL006), the Fundamental Research Fund for the Central Universities (no. Y0201100265), and the foundation of Scientific Research Project of Weifang (no. 20121103).