Two novel energy dissipative difference schemes for the strongly coupled nonlinear space fractional wave equations with damping

Highlights

• Unconditionally convergent energy dissipative numerical methods are developed and analyzed for nonlinear wave equations.

• Two novel efficient energy dissipative difference schemes for the wave equations are first set forth and analyzed.

• The discrete energy dissipation properties, solvability, unconditional convergence and stability results are proven rigidly.

• Some numerical results are provided to illustrate the computational accuracy and efficiency of the proposed schemes.

Abstract

In this paper, two new efficient energy dissipative difference schemes for the strongly coupled nonlinear damped space fractional wave equations are first set forth and analyzed, which involve a two-level nonlinear difference scheme, and a three-level linear difference scheme based on invariant energy quadratization formulation. Then the discrete energy dissipation properties, solvability, unconditional convergence and stability of the proposed schemes are exhibited rigidly. By the discrete energy analysis method, it is rigidly shown that the proposed schemes achieve the unconditional convergence rates of $\mathcal{O}(\mathrm{\Delta }{t}^2+h^2)$ in the discrete $L^{\infty }$-norm for the associated numerical solutions. At last, some numerical results are provided to illustrate the dynamical behaviors of the damping terms and unconditional energy stability of the suggested schemes, and testify the efficiency of theoretical results.

Introduction

In this paper, we concentrate on devising efficient energy dissipative finite difference schemes for solving a class of strongly coupled nonlinear fractional initial boundary value problems (IBVPs) as follows

$${\rho }_1\frac{{\partial }^{2}u}{\partial t^2}+{\kappa }_1{(-\mathrm{\Delta })}^{\frac{\alpha }{2}}u+{\beta }_1{(-\mathrm{\Delta })}^{\frac{\alpha }{2}}\frac{\partial u}{\partial t}+{\beta }_2\frac{\partial u}{\partial t}+F_u(u,v)=0,x\in \mathbb{R},t\in [0,T],$$

$${\rho }_2\frac{{\partial }^{2}v}{\partial t^2}+{\kappa }_2{(-\mathrm{\Delta })}^{\frac{\beta }{2}}v+{\gamma }_1{(-\mathrm{\Delta })}^{\frac{\beta }{2}}\frac{\partial v}{\partial t}+{\gamma }_2\frac{\partial v}{\partial t}+F_v(u,v)=0,x\in \mathbb{R},t\in [0,T],$$

$$u(x,0)=u_0(x),\frac{\partial u}{\partial t}(x,0)=u_1(x),v(x,0)=v_0(x),\frac{\partial v}{\partial t}(x,0)=v_1(x),x\in \mathbb{R},$$

$$u(x,t)=0,v(x,t)=0,x\in \mathbb{R}\backslash \mathrm{\Omega },\mathrm{\Omega }=(a,b),t\in [0,T],$$

where $u(x,t)$ and $v(x,t)$ are real-valued functions, $1<\alpha ,\beta \le 2$, ${\kappa }_1$ and ${\kappa }_2$ are positive diffusivity coefficients. ${\beta }_1$, ${\beta }_2$, ${\gamma }_1$, ${\gamma }_2$ are nonnegative damping parameters and ${\rho }_1$, ${\rho }_2$ are positive parameters. Here $F_u$ and $F_v$ represent the partial derivatives of a nonnegative potential function $F(u,v)$ with respect to u, v, respectively, i.e., $F_u=\frac{\partial F}{\partial u}$, $F_v=\frac{\partial F}{\partial v}$. Moreover, we shall assume that smooth functions u, v, $u_0$, $v_0$, $u_1$ and $v_1$ are spatially compactly supported and satisfy the associated compatibility conditions. The fractional Laplacian ${(-\mathrm{\Delta })}^{\frac{\alpha }{2}}u$ is equivalent to the following Riesz fractional derivative

$$-{(-\mathrm{\Delta })}^{\frac{\alpha }{2}}u(x,t)=\frac{{\partial }^{\alpha }u(x,t)}{\partial |x{|}^{\alpha }}=-\frac{1}{2\cos (\frac{\pi \alpha }{2})}[{}_{-\infty }D_x^{\alpha }u(x,t){+}_x{D}_{+\infty }^{\alpha }u(x,t)],$$

where ${}_{-\infty }D_x^{\alpha }u(x,t)$ and ${}_{x}D_{+\infty }^{\alpha }u(x,t)$ are the left and right Riemann-Liouville fractional derivatives, respectively (cf. [32], [49]). Concerning the numerical approximations of Riesz fractional derivative, a rather incomplete list of numerical methods can be referred to [5], [17], [18], [22], [23], [37], [51] and the related references therein.

Some special cases ($\alpha =\beta =2$) of IBVPs (1a)–(1d) have been broadly explored in physics, chemistry, biology, microstructures, microtechnology and mathematical field [14], [15], [30], [34], [48]. For instance, as $\alpha =\beta =2$ and ${\beta }_1={\beta }_2={\gamma }_1={\gamma }_2=0$, this type of coupled nonlinear space fractional wave equations reduce to the coupled Klein-Gordon system, which can be exploited to describe the evolves of charged mesons in an electromagnetic field [14], [34], [48]. In the last few decades, fractional derivatives have been introduced to mathematical models to elucidate more realistic descriptions of various complex physical phenomena (cf. [32], [36], [49]). This kind of nonlinear coupled space fractional wave equations (1a)–(1d) has sought significant applications in microstructures, chemistry, nonlocal dynamics and biology fields (see [15], [27], [36], [46]). In particular, it can be viewed as the fractional extensions of classical strongly damped nonlinear wave equations [12], [13], [15], [16], [46], [48]. Over the years, analytical studies like global (local) existence and asymptotic behavior of the solution for the coupled nonlinear damped space fractional wave equations have attracted some attention. For example, Pham et al. [31] studied the global existence of small data solutions for semi-linear structurally damped σ-evolution models. D'Abbicco [10] found the critical exponent for global small data solutions to the Cauchy problem in ${\mathbb{R}}^n$, for structurally damped semi-linear space fractional evolution equations. More recently, Dao [13] proved the global (in time) existence of small data energy solutions to the Cauchy problems for a weakly coupled system of semi-linear visco-elastic damped σ-evolution models. More recent research on the well-posedness of wave equation or fractional order equation can refer to the literature [9], [19], [20], [26], [29] and the references therein.

It is remarked that the IBVPs (1a)–(1d) satisfy the following energy dissipative law

$$\frac{\mathrm{d}E}{\mathrm{d}t}=-\underset{\mathbb{R}}{\int }[{\beta }_1{({(-\mathrm{\Delta })}^{\frac{\alpha }{4}}\frac{\partial u}{\partial t})}^2+{\beta }_2{(\frac{\partial u}{\partial t})}^2+{\gamma }_1{({(-\mathrm{\Delta })}^{\frac{\beta }{4}}\frac{\partial v}{\partial t})}^2+{\gamma }_2{(\frac{\partial v}{\partial t})}^2]\mathrm{d}x,t\in (0,T],$$

where

$$E(t)=\underset{\mathbb{R}}{\int }[\frac{{\rho }_1}{2}{(\frac{\partial u}{\partial t})}^2+\frac{{\kappa }_1}{2}{({(-\mathrm{\Delta })}^{\frac{\alpha }{4}}u)}^2+\frac{{\rho }_2}{2}{(\frac{\partial v}{\partial t})}^2+\frac{{\kappa }_2}{2}{({(-\mathrm{\Delta })}^{\frac{\beta }{4}}v)}^2+F(u,v)]\mathrm{d}x,t\in [0,T],$$

and $({(-\mathrm{\Delta })}^{\frac{\alpha }{2}}u,u)=({(-\mathrm{\Delta })}^{\frac{\alpha }{4}}u,{(-\mathrm{\Delta })}^{\frac{\alpha }{4}}u)$. It is crucial to devise high-performance numerical methods which could conserve the intrinsic features of the original system [2], [3], [4], [6], [7], [8], [11], [39], [40], [41], [42], [43] in terms of their superior in stability, long time simulations, and so on. As we know, it is extremely difficult and significant to develop energy dissipative/conservative numerical methods for solving IBVPs (1a)–(1d) because of the presence of strongly coupled nonlinear functions $F_u$ and $F_v$. More recently, Deng and Liang [14] creatively constructed two energy-preserving nonlinear finite difference methods for solving system of nonlinear wave equations ($\alpha =\beta =2$, ${\beta }_1={\beta }_2={\gamma }_1={\gamma }_2=0$) in two dimensions and exhibited the rigid error estimates. The proposed schemes are fairly efficient and perform well for long time simulations. With the development of energy-preserving numerical methods [14], [48] for this kind of nonlinear coupled wave equations ($\alpha =\beta =2$, ${\beta }_1={\beta }_2={\gamma }_1={\gamma }_2=0$), it is interesting and important to develop the efficient numerical methods for solving the coupled nonlinear space fractional wave equation in that the associated fractional model can simulate the more complex physical phenomena. Over the past few years, some energy conservative/dissipative numerical methods [24], [27], [33], [44], [45], [46], [47] have been presented for resolving the Riesz space fractional partial differential equations, together with the proofs of the consistency, convergence and stability. Recently, Macías-Díaz [28] presented a fully explicit scheme for solving the coupled nonlinear fractional hyperbolic equations (1a)–(1d) in the absence of internal damping terms $({\beta }_1={\gamma }_1=0)$ and considered the existence of Turing patterns. This scheme is expected to be easy-to-implement and efficient to some extent. However, the scheme requires the severe restriction condition for the stability and convergence analyses, i.e., “$C\mathrm{\Delta }{t}^2{h}^{1-\alpha }\le 1$” in page 10 of [28]. And moreover, the proposed scheme does not satisfy the discrete energy conservation (${\beta }_1={\beta }_2={\gamma }_1={\gamma }_2=0$) or dissipation property. Therefore, it is meaningful to develop accurate, efficient and easy-to-implement numerical methods which can assure the energy dissipation/conservation property for simulating propagation of the strongly coupled nonlinear space fractional wave equations with damping in long time integration.

In this paper, we aim to devise and analyze two novel types of efficient energy dissipation-preserving finite difference schemes for solving the IBVPs (1a)–(1d), which include a two-level nonlinear finite difference scheme and a three-level linear finite difference scheme based on invariant energy quadratization approach. The discrete energy dissipation properties, solvability, unconditional convergence and stability results of the suggested finite difference methods are provided rigidly. It is shown that the underlying schemes attain the convergence orders of $\mathcal{O}(\mathrm{\Delta }{t}^2+h^2)$ in the discrete $L^{\infty }$-norm, without any time step restrictions dependent on the spatial mesh size. Besides, for the latter scheme, with induction technique at hand, we prove that the discrete $L^{\infty }$-norm of the numerical solutions to the proposed difference scheme are uniformly bounded. Lastly, some numerical results elucidate the physical effects of the damping terms and unconditional energy stability of the proposed finite difference schemes, and validate the efficiency of the proposed theoretical results.

The outline of this paper is arranged as follows. In Section 2, we first present some denotations and auxiliary lemmas, then we concentrate on the derivation of the energy dissipative difference methods for solving the IBVPs (1a)–(1d), along with the associated discrete energy dissipation properties. Section 3 is devoted to the solvability, unconditional convergence and stability of the proposed schemes. Section 4 focuses on the extensions of our proposed schemes and corresponding analytical results to a system consisting of a finite number of fractional wave equations with damping. Numerical experiments are given in Section 5, followed by conclusions in Section 6.

Throughout this paper, C represents a general positive constant independent of h and Δt, which may take various values in different occasions. We utilize $p\precsim q$ to denote $p\le Cq$.