Trigonometrically fitted multi-step RKN methods for second-order oscillatory initial value problems

doi:10.1016/j.amc.2017.10.036

Applied Mathematics and Computation

Volume 320, 1 March 2018, Pages 740-753

https://doi.org/10.1016/j.amc.2017.10.036 Get rights and content

Abstract

Trigonometrically fitted multi-step Runge–Kutta–Nyström (TFMSRKN) methods for solving numerically oscillatory special second-order initial value problems are introduced. TFMSRKN methods integrate exactly the differential system whose solutions can be expressed as the linear combinations of functions from the set ${\exp (i w t), \exp (- i w t)}$ or equivalently the set {cos (wt), sin (wt)}, where w represents an approximation of the main frequency of the problem. The corresponding order conditions are given and two explicit four-stage TFMSRKN methods with order five are constructed. Stability of the new methods is examined and the regions of stability are depicted. Numerical results show that our new methods are more efficient in comparison with other well-known high quality methods proposed in the scientific literature.

Introduction

In this work we consider the effective numerical integration of the initial value problem of second-order differential equations in the form ${\begin{matrix} y^{''} (t) = f (t, y (t)), t \in [t_{0}, T], \\ y (t_{0}) = y_{0}, y^{'} (t_{0}) = y_{0}^{'}, \end{matrix}$ whose solution has an oscillatory character, where y ∈ R^d, f: [t₀, T] × R^d → R^d is sufficiently differentiable and the first derivative y′ does not appear explicitly. Such a problem often arises in different fields of applied sciences such as celestial mechanics, molecular dynamics, quantum mechanics, spatial semi-discretizations of wave equations, electronics, and so on [1], [2], [3].

Considering the oscillatory feature of the problem (1), some researchers have proposed to develop numerical integrators with frequency-dependent coefficients by the techniques like trigonometrical/exponential fitting (see [4], [5]). Early presentations of these techniques come from Gautschi [6] and Lyche [7]. Since then, a lot of exponentially (or trigonometrically) fitted linear multi-step methods have been proposed and applied to other scientific fields. Recently, in the context of Runge–Kutta–Nyström (RKN) methods, exponentially-fitted and trigonometrically-fitted methods have been considered in [8], [9], [10], [11], [12]. Meanwhile mixed-collocation versions have been introduced by Coleman and Duxbury in [13]. All of these methods integrate exactly second-order systems (1) whose solution can be expressed as linear combination of functions from the set of functions ${\exp (i w t), \exp (- i w t)}$ or equivalently the set {cos (wt), sin (wt)}, (see [14], [15]). At the same time, taking the basic frame of two-step hybrid method given by Coleman [16], exponentially (or trigonometrically) fitted two-step hybrid method attracted a wide spread attention [17], [18], [19], [20], [21].

More recently, Li and Wang [22] proposed a family of multi-step Runge–Kutta–Nyström (MSRKN) methods and derived the corresponding order conditions by using the theory of SN-series. Compared with classical one-step RKN methods, MSRKN methods can reach higher order with the same function evaluations per step. In fact, MSRKN methods form a subclass of general linear methods studied by R. D’Ambrosio et al. [23] recently.

Inspired by the previous work, in this paper, we will extend the idea of trigonometrical fitting to MSRKN methods. The rest of this paper is organized as follows: In Section 2, we restate the general formulation of MSRKN methods for the problems (1). Trigonometrical fitting conditions are presented and algebraic order conditions for trigonometrically fitted MSRKN (TFMSRKN) methods are given. In Section 3, the stability properties are analyzed. With the order conditions, two new explicit trigonometrically fitted MSRKN methods of order five, are constructed in Section 4. In Section 5, numerical experiments are carried out and the numerical results show the robustness of the new methods. Section 6 is concerned with conclusions and discussions.

Section snippets

Trigonometrically fitted multi-step Runge–Kutta–Nyström methods

Multi-step Runge–Kutta–Nyström (MSRKN) methods for the problem (1) are defined by the formulas (see [22]) ${\begin{matrix} Y_{i} = \sum_{k = 1}^{l} p_{i k} y_{n - k + 1} + h \sum_{k = 1}^{l} q_{i k} y_{n - k + 1}^{'} + h^{2} \sum_{j = 1}^{s} a_{i j} f (t_{n} + c_{j} h, Y_{j}), \\ i = 1, \dots, s, \\ y_{n + 1} = \sum_{k = 1}^{l} u_{k} y_{n - k + 1} + h \sum_{k = 1}^{l} v_{k} y_{n - k + 1}^{'} + h^{2} \sum_{i = 1}^{s} {\bar{b}}_{i} f (t_{n} + c_{i} h, Y_{i}), \\ y_{n + 1}^{'} = \sum_{k = 1}^{l} w_{k} y_{n - k + 1}^{'} + h \sum_{i = 1}^{s} b_{i} f (t_{n} + c_{i} h, Y_{i}), \end{matrix}$ where c_i, p_ik, q_ik, a_ij, u_k, v_k, ${\bar{b}}_{i},$ w_k and b_i with $k = 1, \dots, l,$ $i, j = 1, \dots, s$ are all real coefficients. The method (2) can be presented by the Butcher-type tableau $\begin{matrix} \begin{matrix} c & P & Q & A \\ u^{T} & v^{T} & {\bar{b}}^{T} \\ w^{T} & b^{T} \end{matrix} = \begin{matrix} c_{1} & p_{11} & \dots & p_{1 l} & q_{11} & \dots & q_{1 l} & a_{11} & \dots & a_{1 s} \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋱ & ⋮ \\ c_{s} & p_{s 1} & \dots & p_{s l} & q_{s 1} & \dots & q_{s l} & a_{s 1} & \dots \end{matrix} \end{matrix}$

Stability

Applying a TFMSRKN method (2) to the following second-order homogeneous linear test mode [25], [26] $y^{''} (t) = - λ y (t), with λ > 0,$ the resulting recursion relation can also be expressed as $\begin{matrix} U_{n} = M (H, v) U_{n - 1} \end{matrix}$ where $U_{n} = {(y_{n + 1}, y_{n}, \dots, y_{n - k + 2}, h y_{n + 1}^{'}, h y_{n}^{'}, \dots, h y_{n - k + 2}^{'})}^{T}$ and $\begin{matrix} M (H, v) = (\begin{matrix} u^{T} - H^{2} {\bar{b}}^{T} N^{- 1} P & v^{T} - H^{2} {\bar{b}}^{T} N^{- 1} Q \\ \tilde{I} & 0 \\ - H^{2} b^{T} N^{- 1} P & w^{T} - H^{2} b^{T} N^{- 1} Q \\ 0 & \tilde{I} \end{matrix}) . \end{matrix}$ with $H = λ h,$ $N = I + H^{2} A$ and $\tilde{I}$ given by $\tilde{I} = {(\begin{matrix} 1 & 0 & \dots & 0 & 0 \\ 0 & 1 & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & \dots & 1 & 0 \end{matrix})}_{(l - 1) \times l}$ Because the stability matrix M(H, v) depends on the variables H and v, geometrically, the characterization of stability

Construction of explicit methods

In this section, we focus our attentions on the construction of the explicit TFMSRKN methods. We consider the following explicit TFMSRKN methods with FSAL property defined by the table of coefficients $\begin{matrix} \begin{matrix} - 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ c_{3} & p_{31} & p_{32} & q_{31} & q_{32} & a_{31} & a_{32} & 0 & 0 \\ 1 & u_{1} & u_{2} & v_{1} & v_{2} & {\bar{b}}_{1} & {\bar{b}}_{2} & {\bar{b}}_{3} & 0 \\ u_{1} & u_{2} & v_{1} & v_{2} & {\bar{b}}_{1} & {\bar{b}}_{2} & {\bar{b}}_{3} & 0 \\ w_{1} & w_{2} & b_{1} & b_{2} & b_{3} & b_{4} \end{matrix} \end{matrix}$ For the coefficients of this method, the conditions (3) give the following fundamental relations $\begin{matrix} p_{31} + p_{32} & = & 1, u_{1} + u_{2} = 1, w_{1} + w_{2} = 1, \\ - p_{32} + q_{31} + q_{32} = c_{3}, - u_{2} + v_{1} + v_{2} = 1 . \end{matrix}$ Under the first two simplifying conditions of (13),

Numerical experiments

In this section, in order to show the competence and superiority of the new methods compared with the well-known methods in the scientific literature, we use four model problems. The criterion used in the numerical comparisons is the decimal logarithm of the global error (GE) versus the computational effort measured in the decimal logarithm of the number of function evaluations required by each method. The integrators we select for comparison are

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L1TFMSRKN2L4S5P: the four-stage

Conclusions and discussions

This paper proposes and studies the trigonometrically fitted multi-step RKN (TFMSRKN) methods for solving oscillatory initial value problem (1). We derived the necessary and sufficient conditions for an TFMSRKN method to be of algebraic order p via the theory of SN-series. Based on the order conditions, two new explicit two-step four-stage methods of order five were constructed. The numerical experiments carried out with several oscillatory problems show that for the problem with oscillatory

Acknowledgements

The authors are grateful to the two anonymous reviewers for their valuable suggestions, which help improve this paper significantly.

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^☆: The research was supported in part by the National Natural Science Foundation of China under Grant No: 11401164, by Hebei Natural Science Foundation of China under Grant No: A2014205136, by the Natural Science Foundation of China under Grant No: 11201113 and and by the Specialized Research Foundation for the Doctoral Program of Higher Education under Grant No: 20121303120001

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Trigonometrically fitted multi-step RKN methods for second-order oscillatory initial value problems☆

Abstract

Introduction

Section snippets

Trigonometrically fitted multi-step Runge–Kutta–Nyström methods

Stability

Construction of explicit methods

Numerical experiments

Conclusions and discussions

Acknowledgements

J. Comput. Appl. Math.

J. Comput. Appl. Math.

Appl. Numer. Math.

Comput. Phys. Commun.

J. Comput. Appl. Math.

J. Comput. Appl. Math.

Comput. Phys. Commun.

J. Comput. Appl. Math.

Comput. Phys. Commun.

Appl. Numer. Math.

J. Comput. Appl. Math.

J. Comput. Appl. Math.

J. Comput. Appl. Math.

Appl. Math. Lett.

J. Comput. Appl. Math.

Numerical solution of highly oscillatory ordinary differential equations

Acta Numer.

Diagonally implicit Rungez–Kuttaz–Nyström methods for oscillatory problems

SIAM J. Numer. Anal.

Solving Ordinary Differential Equations I: Nonstiff Problems