Original article
Trigonometrically fitted two-step hybrid methods for special second order ordinary differential equations

https://doi.org/10.1016/j.matcom.2010.10.011Get rights and content

Abstract

The purpose of this paper is to derive two-step hybrid methods for second order ordinary differential equations with oscillatory or periodic solutions. We show the constructive technique of methods based on trigonometric and mixed polynomial fitting and consider the linear stability analysis of such methods. We then carry out some numerical experiments underlining the properties of the derived classes of methods.

Introduction

It is the aim of our paper to derive new classes of numerical methods for solving initial value problems based on second order ordinary differential equations (ODEs)y=f(x,y),y(x0)=y0,y(x0)=y0with f smooth enough in order to ensure the existence and uniqueness of the solution. Although problem (1) can be solved by transforming it into a system of first order ODEs of double dimension, the development of numerical methods for its direct integration seems more natural and efficient. This problem, having periodic or oscillatory solutions, often appears in many applications: celestial mechanics, seismology, molecular dynamics, and so on (see for instance [23], [26] and references therein contained). Classical numerical methods for ODEs relied on polynomials may not be very well-suited to periodic or oscillatory behaviour. In the framework of exponential fitting many numerical methods have been adapted in order to exactly integrate basis of functions other than polynomials, for instance the exponential basis (see [16] and references therein contained), in order to catch the oscillatory behaviour. The parameters of these methods depend on the values of frequencies, which appear in the solution. In order to adapt the collocation technique [14], [18] to an oscillatory behaviour, the collocation function has been chosen as a linear combination of trigonometric functions [20] or of powers and exponential functions [4]. Many modifications of classical methods have been presented in the literature for problem (1): exponentially fitted Runge–Kutta methods (see for example [13], [24]), or trigonometrically fitted Numerov methods [12], [25] and many others (for a more extensive bibliography see [16] and references in the already cited papers).

The methods we show in this context belong to the class of two-step hybrid methodsYi[n]=(1+ci)ynciyn1+h2j=1saijf(Yj[n]),i=1,,syn+1=2ynyn1+h2i=1sbif(Yi[n]).introduced by Coleman in [3], which can also be represented through the Butcher arraywith c = (c1, c2, …, cs)T, A=(aij)i,j=1s, b = (b1, b2, …, bs)T, where s is the number of stages.

The aim of this paper is to adapt the coefficients of methods (2), (3) to an oscillatory behaviour, in such a way that it exactly integrates linear combinations of power and trigonometric functions depending on one and two frequencies, which we suppose can be estimated in advance. Frequency-dependent methods within the class (2), (3) have already been considered in [28], where the coefficients of methods were modified to produce phase-fitted and amplification-fitted methods.

In Section 2 we rewrite the hybrid method (2), (3) as an A-method, following the idea in [1], [19], in order to regard it as a generalized linear multistep method and consider linear operators associated to it, which will play a crucial rule in the development of the new methods. In Section 3 we derive the methods, by imposing that the internal and external stages exactly integrate linear combinations of mixed basis functions. In particular, we construct methods with constant coefficients and methods with parameters depending on one or two frequencies. In Section 4 we analyze linear stability properties of the derived methods. Finally Section 5 provides numerical tests, in order to illustrate features of the methods and to compare our methods with other ones already known in literature. The paper concludes with an appendix, where we report arrays of some methods.

Section snippets

Two-step hybrid methods as A-methods: order conditions

In this section we show some preliminary and helpful results we will use in the remainder of this paper in order to carry out the construction of numerical methods belonging to the class of two-step hybrid methods (2), (3).

In particular, following the approach introduced by Albrecht (cfr. [1], [19]), we rewrite the class of two-step hybrid methods as A-methods. We first define the following vectors in Rs+2Yn+1=[Y1[n],,Ys[n],yn,yn+1]T,F(xn,Yn+1;h)=[f(xn+c1h,Y1),,f(xn+csh,Ys),f(xn,yn),f(xn,yn+1)

Constructive technique of mixed-trigonometrically fitted two-step hybrid methods

In this section we show the construction of some two-step hybrid methods for the numerical solution of second order ODEs, whose solutions depend on one or more frequencies, which at the moment we suppose can be estimated in advance. In particular, we require that both the internal and external stages of the resulting methods exactly integrate linear combinations of the following basis functions:{1,x,,xq,cos(ωix),sin(ωix),q,i=1,2,}depending on the frequencies ωi, with i, q such that the

Linear stability analysis

We handle the linear stability analysis [21], [26], [27] of the obtained methods. We consider both the cases of methods with constant coefficients and with coefficients depending on one or two frequencies.

Numerical experiments

We now show some numerical results we have obtained applying our families of solvers to some linear and nonlinear problems depending on one or two frequencies, in order to test the accuracy of the derived methods and also to compare them with ones already considered in literature for second order ODEs. Test 1. We consider the following test equationy(x)=25y(x),x[0,2π],y(0)=y0,y(0)=1whose exact solution is y(x) = cos  (5x), so it depends on the frequency ω = 5. We solve this problem using the

Conclusions

In this paper, we present new trigonometrically fitted hybrid methods with parameters depending on one and two frequencies, by modifying the classical hybrid method presented in [3], and analyze the linear stability properties. We think that the used technique can be extended to adapt the coefficients of general linear methods [17] to an oscillatory behaviour, especially in the context of collocation methods, by modifying the choice of the collocation functions, considering not only the

References (27)

  • J.P. Coleman et al.

    P-stability and exponential-fitting methods for y = f(x, y)

    IMA J. Numer. Anal.

    (1996)
  • J.P. Coleman et al.

    Truncation errors in exponential fitting for oscillatory problems

    SIAM J. Numer. Anal.

    (2006)
  • R. D’Ambrosio et al.

    A general family of two step collocation methods for ordinary differential equations

  • Cited by (37)

    • Numerical solution of reaction-diffusion systems of λ-ω Type by trigonometrically fitted methods

      2016, Journal of Computational and Applied Mathematics
      Citation Excerpt :

      For instance, cell cycles are frequently clock-like [2,3], behaving if they are driven by an autonomous biochemical oscillator. Indeed, the periodic character of the problem suggests to propose a numerical solution of (1.2) which takes into account this qualitative behavior, i.e. by means of a special purpose numerical solver more tuned to follow the periodic behavior, in the spirit of the so-called exponential fitting technique (EF, refer to the recent review paper on the topic [10] and references therein and the classical monograph [11]; in the case of differential equations, we specifically refer to [12–19] and references therein). The existing literature on EF-based methods has provided a certain number of adaptations of classical numerical methods to better numerically follow known qualitative behaviors (e.g. periodicity, oscillations, exponential decay of the solution).

    • Functionally-fitted explicit pseudo two-step Runge-Kutta-Nyström methods

      2015, Applied Numerical Mathematics
      Citation Excerpt :

      These methods are developed to integrate an ODE exactly if the solution of the ODE is a linear combination of some certain basis functions (see, e.g., [2,7–10,13–16,19–21,25,26,28]). For example, trigonometric methods have been developed to solve periodic or nearly periodic problems (see, e.g., [7,15,18,26,27]). Numerical experiments have shown that trigonometrically-fitted methods are superior to classical Runge–Kutta methods for solving ODEs whose solutions are periodic or nearly periodic functions with known frequencies (see, e.g., [13,18–21,24,26]).

    • An eight-step semi-embedded predictor-corrector method for orbital problems and related IVPs with oscillatory solutions for which the frequency is unknown

      2015, Journal of Computational and Applied Mathematics
      Citation Excerpt :

      Second-order ordinary differential equations are very important in many areas of quantum mechanics, quantum chemistry, physical chemistry and chemical physics, astrophysics, astronomy, celestial mechanics or electronics. There are interesting and efficient methods in the literature for the solution of the problems of the form (1), Chebyshev-based exponentially-fitted methods (see [4]), Legendre–Gauss collocation methods (see [5]), trigonometrically-fitted hybrid methods (see [6]), linear symmetric multistep methods (see [7]), phase-fitted predictor–corrector methods (see [8]). The methods for the solution of the problems of the form (1) are divided into two categories:

    • Ef-Gaussian direct quadrature methods for Volterra integral equations with periodic solution

      2015, Mathematics and Computers in Simulation
      Citation Excerpt :

      Much more recently a Direct Quadrature (DQ) method has been proposed, which is based on an exponentially fitted (ef) quadrature rule of Simpson type [12], and the reason behind this way of looking at the things was that the exponential fitting is an approach specially devised to work on periodic functions. Also, the ef formalism is extremely flexible to cover a large diversity of numerical operations including interpolation, quadrature and numerical solution of ordinary differential equations (ODEs) [26], and massive experimental evidence has been accumulated along time that the ef-based methods perform much better than classical methods, see, e.g., [16,19–23,26,27,32] and the monograph [28]. The quadrature rule used in [12] was the ef-based Simpson rule but in this paper we go one step further.

    View all citing articles on Scopus
    View full text