Dissipation-preserving Galerkin–Legendre spectral methods for two-dimensional fractional nonlinear wave equations☆
Introduction
A class of nonlinear damped and undamped wave equations, i.e., sine–Gordon-like equation and Klein–Gordon-like equation, are extensively described in nonlinear optics, supertransmission, hydrology, anomalous diffusion and other phenomenons [1]. The nonlinear undamped wave equation is as follows [2], [3], [4], In addition, (1.1) contains the conservation of energy where is nonlinear and nonnegative, and is bounded. In particular, if , (1.1) can reduce to sine–Gordon equation; if , (1.1) becomes Klein–Gordon equation.
In recent decades, the investigation of nonlinear wave equations including the nonlocal effect was considered. The fractional models were widely applied in science and engineering which well describe the long-range interaction, such as the interaction of solitons in a collisionless plasma and the presence of the phenomenon of nonlinear supratransmission of energy [5]. In this paper, we consider dissipation-preserving Galerkin–Legendre spectral methods to solve the following fractional wave equations in two dimensions: where , the parameters and , and is the parameter of damping force, . is nonlinear, which satisfies and the assumption that is bounded. Denote and as the left and right Riemann–Liouville fractional derivatives defined on bounded domain , respectively. In addition, and can be defined similarly [6]. At the same time, and are the Riesz fractional operators defined by where and . We define and . The nonlinear fractional wave model possesses energy function , and the energy function from (1.3)–(1.5) holds that If , the energy is conserved. If , the energy is dissipative.
In recent years, the space fractional nonlinear wave equation appears in many physical fields, and it covers a series of fractional differential equations, such as the space fractional telegraph equation [7], the space fractional Klein–Gordon equation [8], the space fractional sine–Gordon equation [9]. Since it is difficult to obtain the analytical solution of fractional nonlinear partial differential equation, the numerical methods for space fractional differential equations get rapid development, such as finite difference methods [10], [11], [12], [13], [14], [15], spectral methods [16], [17], [18], [19], [20] and finite element methods [21], [22], [23]. In addition, the fractional nonlinear wave equation possesses the dissipation or conservation of the energy. Therefore, the numerical methods preserving the intrinsic property are very important. Recently, the structure-preserving methods have attracted more and more researchers’ attention [24], [25], [26], [27], [28], [29]. Motivated by the consideration, some works focused on constructing structure-preserving finite difference methods for the space fractional nonlinear wave equation. Macías-Díaz et al. [30], [31], [32] proposed a series of efficient structure-preserving finite difference methods to solve the fractional sine–Gordon equation with Riesz fractional derivative, which get the convergence order of second in space. Furthermore, they [33] also proposed a energy-preserving compact finite difference method to solve the fractional nonlinear wave equations, which gets convergence order of fourth in space. Recently, Xie et al. [34] developed an efficient dissipation-preserving fourth-order difference method for the space fractional nonlinear wave equations in one dimension, and the stability and convergence were proved in detail. Due to the non-local property of fractional derivative, spectral method as global method is natural choice and will enjoy the advantage of high accuracy. However, up to now, there are no works on structure-preserving spectral method for the fractional nonlinear wave equation, let alone the high dimensional case. In this work, we use the same finite difference method as [34] in time discretization, while the spectral method is applied in space discretization for two dimensional case to get the high accuracy. In addition, we assume that the nonlinear function is bounded in suitable domain as the assumption in [16], which is a weaker condition than [33], [34]. Meanwhile, the stability and convergence analysis are given under the weaker condition. For the two dimensional case, the matrix diagonalization method can efficiently solve resulting algebraic system of the spectral method without introducing the small term, while the alternating direction method needs.
The main contribution of this paper is to develop two dissipation-preserving Galerkin–Legendre spectral methods for solving the fractional nonlinear wave equation, i.e., the Crank–Nicolson Galerkin–Legendre spectral method (CNGLS) and the three-level linearized Galerkin–Legendre spectral method (LGLS). The two methods preserve the dissipation of energy for , and the conservation of energy for . The stability and convergence analysis are given under the weaker condition, which show that the CNGLS and LGLS methods are both conditionally stable. In addition, the two methods are convergent with second order accuracy in time and optimal error estimates in space. In numerical implementation, in order to reduce the computing cost, we adopt the matrix diagonalization method to solve the resulting algebraic systems of implicit and linearized full discrete schemes. Numerical experiments are provided to confirm the theoretical results and validate the efficiency of our algorithms.
The rest of paper is organized as follows. In Section 2, we recall some technical lemmas and notations. In Section 3, we propose the CNGLS method and show the CNGLS scheme preserves energy dissipation. Moreover, the stability and convergence of the CNGLS method are strictly proven, and the implementation of the CNGLS scheme by using matrix diagonalization method is given in detail. In Section 4, the LGLS method and the corresponding discrete energy dissipation-preserving structure are established. The stability and convergence analysis of the LGLS method are provided. In addition, the step by step algorithm of the LGLS method is established in detail. The numerical experiments are performed to confirm the correctness of theoretical analysis in Section 5. In the end, some conclusions are drawn in Section 6.
Section snippets
Preliminaries
In this section, we first recall some notations, definitions and lemmas which play an important role in subsequently theoretical analysis.
Crank–Nicolson implicit Galerkin–Legendre spectral method
In this section, we first present the CNGLS method to solve (1.3)–(1.5). In addition, the dissipation or conservation of energy is established. We also give detailed stability and convergence analysis, and the implementation of the algorithms is provided in detail.
Linearized Galerkin-Legendre spectral method
In above section, we consider the CNGLS method to solve the space fractional nonlinear wave equation, however the scheme is full implicit and needs iteration method to solve the algebraic system at each time step. The linearized method only needs to solve a linear system at each time level. Thus, we also propose a linearized spectral scheme to solve the space fractional nonlinear wave equation.
First, the variational formulation for (1.3)–(1.5) is following: Find such that
Numerical experiments
In this section, we present some numerical results of the CNGLS and LGLS methods to confirm our theoretical analysis. To reduce computational cost, we adopt matrix diagonalization (MDA) method to solve the resulting algebraic systems for our methods. All experiments were performed on a windows10 (64 bit) PC-Intel (R) Core (TM) i7-8650 CPU 1.90 GHz, 16 GB of RAM using MATLAB 2017a.
First, we will verify the error estimates for the CNGLS and LGLS methods. For simplicity, here we just consider the
Conclusion
In this paper, we have constructed dissipation-preserving Crank–Nicolson and linearized Galerkin–Legendre spectral methods to solve the fractional nonlinear wave equations in two dimensions. Both methods preserve the corresponding energy dissipation and energy conservation as continuous model. Moreover, we also give the detailed derivation of full discrete energy structures and convergence analysis for the two schemes. In addition, to reduce the computing cost, we adopt the matrix
CRediT authorship contribution statement
Nan Wang: Conceptualization, Methodology, Formal analysis, Software, Validation, Investigation, Resources, Data curation, Writing - original draft, Visualization. Mingfa Fei: Conceptualization, Methodology, Formal analysis, Investigation, Validation, Writing - review & editing, Visualization, Supervision, Project administration, Funding acquisition. Chengming Huang: Conceptualization, Methodology, Validation, Formal analysis, Writing - review & editing. Guoyu Zhang: Conceptualization,
Acknowledgments
The authors wish to thank the anonymous referees for their valuable comments and suggestions to improve this paper.
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This work was supported in part by NSF of China (11771163, 11801527), China Postdoctoral Science Foundation (2019M662506), Scientific Research Fund of Hunan Provincial Education Department, China (19C0177, 19C0181) and the Hunan Province Key Laboratory of Industrial Internet Technology and Security, China (2019TP1011).