A fourth-order dissipation-preserving algorithm with fast implementation for space fractional nonlinear damped wave equations

Highlights

• We propose a new fourth-order dissipation-preserving difference scheme for space fractional nonlinear damped wave equations.

• Our proposed fourth-order scheme can generate the full toeplitz matrix, it is convenient for us to speed up the matrix-vector multiplication by fast Fourier transform.

• We obtain the L² and H^α/2 norm estimates, which are discussed for two-dimensional (high-dimensional) space fractional nonlinear damped wave equations.

• We obtain the convergence analysis of fully-discrete scheme without any restriction on step-size ratio.

• Our proposed scheme performs more robust than some existing schemes for conserving dissipation-preserving law.

Abstract

In this paper, we numerically investigate the space fractional nonlinear damped wave equation. We construct a novel high-accuracy dissipation-preserving finite difference scheme by using the new fourth-order fractional central difference operator. Thanks to the toeplitz-like differentiation matrix, we further raise the computation efficiency of the proposed scheme by fast Fourier transform. Moreover, we obtain the error estimate of our proposed scheme in L² and H^α/2 (1 < α ≤ 2) norm, respectively. Finally, we verify the discrete dissipation-preserving law and convergence of the proposed scheme by one- and two-dimensional numerical experiments in long-time observation.

Introduction

In the past decades, fractional calculus frequently appear in many fields, such as physics, biomedicine, image processing and quantum mechanics, etc. More and more mathematicians, physicists and engineers recognize that fractional partial differential equations possess the capacity to depict complex and particular non-local phenomena or process with memory and long-range interaction. Now, some fractional differential models, such as, fractional Schrödinger equation [23], [24], [25], fractional Ginzburg-Landau equation [42], and fractional sine-Gordon equation [2] etc. have been proposed from classical differential equations by replacing the integral derivative with fractional derivative. A great number of simulation results (see [15], [16], [45], [46] and references therein) reveal that these attempts are necessary and successful.

In recent researches, fractional nonlinear wave equation gains more and more attentions in constructing numerical scheme. It contains a series of important fractional models, such as fractional telegraph equation [8], fractional Cattaneo equation [37], fractional Kelin-Gordon equation [17], [19], and fractional sine-Gordon equation [2]. Many nonlinear differential models possess some physics quantities, such as energy, momentum and mass, which are preserved under suitable boundary conditions. Thus, designing the structure-preserving numerical scheme for fractional nonlinear wave equation is an important topic. For integral nonlinear wave equation, the numerical and relevant theoretical aspects have been studied [9], [10], [11], [12], [22], [39]. But for fractional nonlinear wave equation, it is difficult to construct the numerical scheme and obtain the corresponding theoretical analysis due to the non-local property of fractional derivative. The analytical solutions of nonlinear wave equations are also investigated, which can be referred to references [2], [5], [30], [35].

In this paper, we consider the following space fractional nonlinear damped wave equation (SFWE)

$$\frac{{\partial }^{2}u}{\partial t^2}(\mathit{x},t)+\gamma \frac{\partial u}{\partial t}(\mathit{x},t)-L^{\alpha }u(\mathit{x},t)+G^{\prime }\left(u(\mathit{x},t)\right)=0,\mathit{x}\in {\mathbb{R}}^d(d\ge 1),0<t\le T,$$

subject to the initial conditions

$$u(\mathit{x},0)=\phi \left(\mathit{x}\right),u_t(\mathit{x},0)=\psi \left(\mathit{x}\right),\mathit{x}\in {\mathbb{R}}^d,$$

and the boundary conditions

$$u(\mathit{x},t)=0,\mathit{x}\in {\mathbb{R}}^d\setminus \Omega ,0<t\le T,$$

where 1 < α ≤ 2, and γ ≥ 0, $G^{\prime }\left(u\right)=\frac{dG}{du}$. We assume that u, φ, ψ are spatially compactly supported on the bounded domain Ω. Besides, we postulate that the given functions φ, ψ are known smooth enough and the nonnegative function G are differentiable with respect to u. So that our proposed scheme can attain the desired convergence order.

If $d=1,$ then

$$L^{\alpha }=\frac{{\partial }^{\alpha }}{{\partial \left|x\right|}^{\alpha }},x\in \mathbb{R}.$$

If $d=2,$ then

$$L^{\alpha }=\frac{{\partial }^{\alpha }}{{\partial \left|x\right|}^{\alpha }}+\frac{{\partial }^{\alpha }}{{\partial \left|y\right|}^{\alpha }},(x,y)\in {\mathbb{R}}^2.$$

The Riesz fractional derivative defined by

$$\frac{{\partial }^{\alpha }}{{\partial \left|x\right|}^{\alpha }}u\left(x\right)=c_{\alpha }({}_{-\infty }^{RL}{D}_x^{\alpha }+{}_x^{RL}{D}_{\infty }^{\alpha })u\left(x\right),c_{\alpha }=-\frac{1}{2\cos \left(\frac{\alpha \pi }{2}\right)},$$

the left and right Riemann-Liouville fractional derivatives are defined as

$${}_{-\infty }^{RL}{D}_x^{\alpha }u\left(x\right)=\frac{1}{\Gamma (2-\alpha )}\frac{d^2}{d{x}^2}{\int }_{-\infty }^x\frac{u\left(\xi \right)d\xi }{{(x-\xi )}^{\alpha -1}},{}_x^{RL}{D}_{\infty }^{\alpha }u\left(x\right)=\frac{1}{\Gamma (2-\alpha )}\frac{d^2}{d{x}^2}{\int }_x^{\infty }\frac{u\left(\xi \right)d\xi }{{(\xi -x)}^{\alpha -1}},$$

where $\Gamma \left(z\right)={\int }_0^{\infty }{x}^{z-1}{e}^{-x}dx$.

As indicated in Laskin [25], Riesz fractional derivative is a self-adjoint and negative operator. Since any positive self-adjoint operator has a unique positive square root [14], there exist unique positive square roots ${\nabla }_x^{\alpha /2}$ and ${\nabla }_y^{\alpha /2},$ such that

$$\underset{{\mathbb{R}}^2}{\int \int }|{\nabla }_x^{\alpha /2}{u|}^{2}dxdy=-\underset{{\mathbb{R}}^2}{\int \int }\left(\frac{{\partial }^{\alpha }u}{{\partial \left|x\right|}^{\alpha }}\right)udxdy,\underset{{\mathbb{R}}^2}{\int \int }{\left|{\nabla }_y^{\alpha /2}u\right|}^{2}dxdy=-\underset{{\mathbb{R}}^2}{\int \int }\left(\frac{{\partial }^{\alpha }u}{{\partial \left|y\right|}^{\alpha }}\right)udxdy.$$

This paper is restricted only to two dimensional (2D) case, that is $d=2$. Thus, the fractional energy¹ formula for (1.1)–(1.3) can be written as

$$\mathcal{E}\left(t\right)=\underset{{\mathbb{R}}^2}{\int \int }[\frac{1}{2}|\frac{\partial u}{\partial t}{|}^2+\frac{1}{2}|{\nabla }_x^{\alpha /2}{u|}^2+\frac{1}{2}{\left|{\nabla }_y^{\alpha /2}u\right|}^2+G\left(u(x,y,t)\right)]dxdy,t\in [0,T],$$

which satisfies

$$\frac{d\mathcal{E}}{dt}=-\gamma \underset{{\mathbb{R}}^2}{\int \int }{\left|\frac{\partial u}{\partial t}\right|}^{2}dxdy,\gamma \ge 0,t\in (0,T].$$

In particular, if $\gamma =0,$ SFWE is energy-conservation, otherwise it is dissipation-preserving.

As far as we know, there are some numerical schemes have been developed in the literatures for SFWE. For instance, Macías-Díaz [31], [33] proposed the second-order accuracy energy-conservation difference schemes for 1D SWFE. Nevertheless, they need a strict assumption in convergence analysis, namely $\frac{{\tau }^2{g}^{\left(\alpha \right)}}{2h^{\alpha -1}}+C\tau <1$ with C a fixed constant independent of mesh-size. Further, the author proposed a fourth-order compact energy-conservation scheme for 1D SFWE when $\gamma =0$ in Macías-Díaz et al. [32]. However, Xiao and Wang [47] affirmed that the matrix $M^{-1}{\delta }_x^{\left(\alpha \right)}$ generated from fourth-order compact operator is not symmetric for α ≠ 2, thus, we believe that the energy-conservation law of fourth-order compact scheme can not be obtained in long-time numerical computation. More recently, Zhao et al. [54] constructed an explicit energy-conservation scheme for 1D SFWE when $\gamma =0,$ which is fourth-order accuracy in time and space, respectively. But the numerical analysis is only discussed for the semi-discrete finite difference scheme. To remedy the shortages of the previous works, we construct a new fourth-order dissipation-preserving difference scheme for SFWE, and we obtain separately the error estimates in L² and H^α/2 norm by rigorous mathematical proof. Also, our proposed scheme performs more excellent than some existing literatures by comparing the corresponding relative energy errors and errors in dissipation-preserving law.

The outline of this paper is arranged as follows. In Section 2, we introduce some high-order approaches of fractional derivative and some useful lemmas. In Section 3, we construct a fourth-order numerical scheme for SFWE, and prove the dissipation-preserving law of fully-discrete scheme. The solvability and convergence analysis of the proposed scheme are discussed in Section 4. In Section 5, we introduce the fast Fourier transform algorithm to speed up the matrix-vector multiplication. Moreover, we provide some numerical examples of 1D and 2D to verify the theoretical analysis of our proposed scheme. The paper ends with a conclusion in Section 6.