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In order to solve initial value problems of differential equations with oscillatory solutions, this paper improves traditional Runge-Kutta (RK) methods by introducing frequency-depending weights in the update. New practical RK integrators are obtained with the phase-fitting and amplification-fitting conditions and algebraic order conditions. Two of the new methods have updates that are also phase-fitted and amplification-fitted. The linear stability and phase properties of the new methods are examined. The results of numerical experiments on physical and biological problems show the robustness and competence of the new methods compared to some highly efficient integrators in the literature.

A vast of problems in applied fields, such as elastics, mechanics, astrophysics, electronics, molecular dynamics, ecology, and biochemistry, can be cast into the form of an initial value problem of the system of first-order ordinary differential equations as follows:

Recently, some authors have proposed to adapt traditional integrators to the oscillatory character of the solution to the problem (

Based on the previous work, we consider, in this paper, phase-fitted and amplification-fitted RK type integrators whose coefficients in the update depend on the product of the fitting frequency and the step size. In Section

Assume that the principal frequency of the problem (

As explained in [

The modified RK type method (

If the method (

For oscillatory problems, as suggested by Paternoster [

An application of the modified RK method (

The quantities

It is interesting to consider the phase properties of the update of the scheme (

The quantities

Generally speaking, a traditional RK method with constant coefficients inevitably carries a nonzero phase lag and a nonzero error of amplification factor when applied to the linear oscillatory equation (

The following theorem gives the necessary and sufficient conditions for a modified RK method and its update to be phase-fitted and amplification-fitted, respectively.

(i) The method (

(ii) The update of the method (

The proof of this theorem is immediate.

Now we proceed to construct modified RK type methods that are both phase-fitted and amplification-fitted based on the internal coefficients of two classical RK methods. For convenience we restrict ourselves to explicit methods.

Consider a four-stage modified RK method with the following Butcher tableau:

Corresponding to the nine fifth-order rooted trees

Now we require that the update of the method (

The error coefficients of FRK4 are given by

It can be seen that as

Consider the following modified RK type method with FSAL property, the prototype of which can be found on page 167 of [

For the method (

By simple computation, we have

By Theorem

Corresponding to the twenty sixth-order rooted trees

Now we require that the update of the method (

It can be verified that the method given by (

The error coefficients of FRK5b are given by

It can be seen that as

In order to investigate the linear stability of the methods (

The region in the

The imaginary stability regions of the methods Simos4, FRK4, FRK5a and FRK5b are depicted in Figures

The imaginary stability region of Simos4 (a) and FRK4a (b).

The imaginary stability region of FRK5a (a) and FRK5b (b).

The quantities

By definition, an FRK method has zero phase lag and zero dissipation when applied to the standard linear oscillator (

Suppose that the internal stages have been exact for the linear equation (

Letting

Simos4:

FRK4:

FRK5a:

FRK5b:

Note that if the update of (

In order to examine the effectiveness of the new FRK methods proposed in this paper, we apply them to four test problems. We also employ several highly efficient integrators from the literature for comparison. The numerical methods we choose for experiments are as follow:

FRK5a: the seven-stage RK method of order five given by (

FRK5b: the seven-stage RK method of order five given by (

Simos4: the RK method of order four presented in [

FRK4: the four-stage RK method of order four given by (

ARK4: the second four-stage adapted RK method of order four given in the Subsection 3.2 of Franco [

EFRK4: the four-stage exponentially fitted RK method of order four given in [

RK4: the classical RK method of order four presented in [

RK5: the classical RK method of order five presented in [

Consider the following orbit problem studied in [

Efficiency curves of Problem

Consider the following linear problem studied in [

Consider the prey-predator system in ecology (see [

In this experiment, we take the values of parameters

Efficiency curves of Problem

We consider the following two-gene regulatory system without self-regulation (see Widder et al. [

The solution

In this experiment, we take initial data

It can be seen from Figures

In this paper, classical Runge-Kutta methods are adapted to the time integration of initial value problems of first order differential equations whose solutions have oscillatory properties. The newly developed phase-fitted and amplification-fitted Runge-Kutta methods (FRK) adopt functions of the product

As the fitting frequency tends to zero, FRK methods reduce to their classical prototypes methods. Furthermore, an FRK method has the same algebraic order and the same error constant with its prototypes method. Numerical experiments illustrate the high efficiency of FRK methods compared with their prototype methods and some other frequency depending methods like exponentially fitted RK type methods.

Theorem

In practical computations of oscillatory problems, the true frequency is, in general, not available. The fitting frequency contained in an FRK method is just an estimate of the true frequency. Sometimes the choice of the value of the fitting frequency

This paper provides a convenient approach to constructing FRK methods. Other effective approaches are possible. For example, the fitting frequency can also be incorporated into internal coefficients

This work was supported by the National Natural Science Foundation of China (Grant no. 11171155), the Fundamental Research Fund for the Central Universities (Grants no. Y0201100265 and KYZ201125), and the Research Fund for the Doctoral Program of Higher Education (Grant no. 20100091110033). The authors would like to thank the referees for their valuable comments and suggestions.