Elsevier

Applied Numerical Mathematics

Volume 119, September 2017, Pages 164-178
Applied Numerical Mathematics

Efficient implementation of RKN-type Fourier collocation methods for second-order differential equations

https://doi.org/10.1016/j.apnum.2017.04.008Get rights and content

Abstract

In this paper we discuss the efficient implementation of RKN-type Fourier collocation methods, which are used when solving second-order differential equations. The proposed implementation relies on an alternative formulation of the methods and their blended formulation. The features and effectiveness of the implementation are confirmed by the performance of the methods on three numerical tests.

Introduction

The efficient numerical solution of implicit methods for differential equations has been the subject of many investigations in the last decades. This paper deals with the numerical solution of second-order differential equations, namely problems in the formq(t)=f(q(t)),q(t0)=q0,q(t0)=q0,t[t0,tend], where q(t):RRd and f(q):RdRd is an analytic function. The solution of this system and its derivative satisfy the following variation-of-constants formula ([30]) with the stepsize h and μ[0,1]:q(t+μh)=q(t)+μhq(t)+h20μ(μx)f(q(t+hx))dx,q(t+μh)=q(t)+h0μf(q(t+hx))dx. Numerical methods of the second-order system (1) have been studied by many researchers in the last decades (see, e.g. [16], [17], [19], [20], [22], [23], [25], [26], [27], [28], [29], [31]), and Runge–Kutta–Nyström (RKN) methods are one of well-known methods for solving these systems.

In [24], the authors took advantage of shifted Legendre polynomials to obtain a local Fourier expansion of the considered system and derived a kind of collocation methods (trigonometric Fourier collocation methods). The trigonometric Fourier collocation (TFC) method for integratingq(t)+Mq(t)=g(q(t)),q(0)=q0,q(0)=p0, is defined asvi=ϕ0(ci2V)q0+cihϕ1(ci2V)p0+(cih)2j=0r1I1,j,ci(V)l=1kblPˆj(cl)g(vl),i=1,2,,k,v(h)=ϕ0(V)q0+hϕ1(V)p0+h2j=0r1I1,j(V)l=1kblPˆj(cl)g(vl),u(h)=hMϕ1(V)q0+ϕ0(V)p0+hj=0r1I2,j(V)l=1kblPˆj(cl)g(vl), where h is the stepsize, Pˆj are the shifted Legendre polynomials over the interval [0,1], and cl,bl,l=1,2,,k are the node points and the quadrature weights of a quadrature formula, respectively. The definitions of ϕ0,ϕ1,I1,j,I2,j,I1,j,ci can be found in [24]. The analysis in [24] demonstrates an important feature, namely that the trigonometric Fourier collocation methods can be of an arbitrary order and when M0, each trigonometric Fourier collocation method creates a particular RKN-type Fourier collocation method, which is symplectic under some conditions. It is noted that these RKN-type Fourier collocation methods are implicit and an iterative procedure for solving the generated discrete problems is required. On the other hand, efficient implementation of implicit methods has been investigated by many researchers in recent years and we refer to [2], [3], [6], [8], [12], [13], [14] for some examples on this topic. Motivated by these publications, this paper presents an efficient implementation of RKN-type Fourier collocation methods, by proposing and analyzing an iterative procedure based on the particular structure of the methods.

With this premise, the paper is organized as follows. Section 2 describes the derivation of RKN-type Fourier collocation methods and the structure of the discrete problem generated by the methods. In Section 3 we propose and analyze an efficient implementation of RKN-type Fourier collocation methods. Section 4 reports some numerical tests to show the features and effectiveness of the methods. The last section contains a few conclusions.

Section snippets

RKN-type Fourier collocation methods

RKN-type Fourier collocation methods are given in [24] as a by-product of trigonometric Fourier collocation methods for second-order differential equations. We now recall the derivation of the methods, and derive the most efficient formulation of the generated discrete problems.

Let us consider the restriction of problem (1) to the interval [t0,t0+h], with the right-hand side expanded along the shifted Legendre polynomials {Pˆj}j=0 over the interval [0,1], scaled in order to be orthonormal (see

Implementation of the methods

In this section, we propose and analyze the efficient implementation of RKN-type Fourier collocation methods.

Numerical tests

In this section, numerical examples are reported to highlight the features and show the effectiveness of the methods. As an example of RKN-type Fourier collocation methods, we choose k=8,r=2 and ci,bi are chosen as the points and quadrature weights of the eight-point Gauss–Legendre's quadrature, respectively. The corresponding method is denoted by RKN-FC(8;2). In order to show the efficiency and robustness of the method, the integrators we select for comparison are:

  • RKN-FC(8;2): the RKN-FC(8;2)

Conclusions

In this paper we propose and analyze an efficient iterative procedure for solving the second-order differential equations (1) generated by the application of RKN-type Fourier collocation methods. The proposed implementation turns out to be robust and efficient. Three numerical tests confirm the effectiveness of the proposed iteration when numerically solving second-order differential equations.

Last but not least, it is noted that there are still some issues which will be further considered.

  • A

Acknowledgements

The authors are sincerely grateful to two anonymous reviewers for their valuable suggestions, which help improve the presentation of the manuscript significantly.

References (31)

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    Recently, similar physical systems such as “Good” Boussinesq equations have been researched by many publications and we refer to [7,9,11,57,59,60] as well as the references contained therein. We also note that much work has appeared in recent years for the research of numerical approximations of Hamiltonian ordinary differential equations (see, e.g. [6,21,24,34,48,49,52,54,55]). Due to the symplectic geometric structure, Hamiltonian systems have important applications in mechanics, celestial and molecular dynamics, and optics.

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This paper was supported by National Natural Science Foundation of China (Grant Nos. 11401333, 11671227, 11571302), by Natural Science Foundation of Shandong Province (Grant No. ZR2014AQ003), by China Postdoctoral Science Foundation (Grant No. 2015M580578), and by Postdoctoral Innovation Project of Shandong Province (Grant No. 201602034), and by Foundation of Scientific Research Project of Shandong Universities (Grant No. J14LI04).

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