Highly efficient difference methods for stochastic space fractional wave equation driven by additive and multiplicative noise

https://doi.org/10.1016/j.aml.2020.106988Get rights and content

Highlights

  • The trace formulas for the energy are derived for two different noises.

  • A new fourth-order scheme is proposed for the stochastic fractional wave equation.

  • The discrete energy of the full discrete scheme is consistent with theoretical results.

  • The scheme preserves dissipation-preserving or energy conservation law.

  • The strong convergence order of the scheme in time is presented.

Abstract

A new fully discrete scheme of stochastic space-fractional nonlinear damped wave equations respectively driven by additive and multiplicative noise is presented. Based on the spatial discretization done via a fourth-order central difference scheme, the semi-implicit Crank–Nicolson scheme is developed for the temporal approximation. The trace formulas for the energy are given for both additive noise and multiplicative Stratonovich noise. Furthermore, the discrete energy trace formula of numerical scheme observed is consistent with behavior of theoretical analysis results. Otherwise, the convergence of order for numerical scheme in time is calculated. Some numerical experiments are performed to verify theory in long time computations at last.

Introduction

In order to better model wave propagation in practical physical problems, general wave equation sometimes needs to add some linear, non-linear, damped, fractional Laplacian terms and stochastic factors, etc.

In the last two decades, stochastic partial differential equations (SPDE) and random partial differential equations [1] have attained more attention in various fields of natural sciences, financial mathematics, engineering, such as in the motion of a DNA molecule in a fluid and the internal structure of the sun [2] including stochastic Schrödinger equation [3], stochastic wave equation [4], stochastic Allen–Cahn equation [5]. In particular, there have been discussions on many real applications for stochastic wave equation driven additive and multiplicative noise. For example, some biological events are related to the motion of the DNA string. The molecular forces are reasonably modeled by a stochastic noise term when the DNA strand moves. Therefore, this phenomenon can be described by stochastic wave equation in [2], [6]. Furthermore, the stochastic fractional wave equation can model wave propagation with the long rang interaction and heterogeneity of media and material in case of various stochastic perturbations from many natural sources [7]. Due to the broad application, it is seldom possible to solve stochastic PDE exactly, some efficient numerical methods for solving SPDE driven by additive and multiplicative noises have been developed, we can refer the interested readers to [8] and the references therein.

More recently, Li and Deng [7] discussed the Galerkin finite element approximations for the stochastic space–time fractional wave equation forced by an additive space–time white noise. As we know, the analytical solutions of SPDE and fractional PDE are difficult to obtain, this motivates us to focus on the numerical methods for stochastic space-fractional nonlinear damped wave equations (SFNDWE) driven by additive and multiplicative noises. In the deterministic case, many efficient numerical methods are proposed for solving Riesz space-fractional nonlinear wave equations, see [9], [10] for more details. To the best of our knowledge, there have been so far a few research results on high-order scheme to solve SPDE. Chen in [11] proposed a compact scheme for stochastic nonlinear Schrödinger equations and established the theory of physical property-preserving. In the current publications, little literature is devoted to SFNDWE of additive and multiplicative noise of ut type. Our work is concerned with the following SFNDWE. dut+[(Δ)α2u+βut+F(u)]dt=λg(ut)dW,u(x,0)=φ(x),ut(x,0)=ψ(x),xR,u(x,t)=0,xRΩ,Ω=(a,b),t[0,T], where (Δ)α2u(x)=12cosαπ2(Dxαu(x)+xD+αu(x)), λ denotes size of noise and the symbol denotes that the product is Stratonovich product and ut=ut. W is an L2(D)-valued Q-Wiener process defined on a given probability space (Ω,F,P) with filtration {Ft}0tT. Let {em}mN be an orthonormal basis of L2(D) that consists of eigenvectors of a symmetric, nonnegative and finite trace operator Q, i.e, Tr(Q)=mNQem,emL2=mNηm< and Qem=ηmem. Then there exists a sequence of independent real-valued Brownian motions {βm}mN such that W(t,x,ω)=m=0ηmβm(t,ω)em(x),ωΩ,xD,t0.

In this article, we derive trace formula (the expected value of the energy) of continuous systems for SFNDWE with additive and multiplicative noise. The key to obtaining preserving-trace formula for the energy is to construct an efficient and easily implemented numerical method. To study qualitatively the solution and energy of (1.1), we use the high resolution and economy method to solve (1.1) in space. On the one hand, we present a full discrete scheme with Crank–Nicolson scheme in time and fourth-order central difference scheme in space. On the other hand, we show that discrete trace formula of energy coincides with theoretical analysis result and preserves structure characteristic of energy. We inspect the behavior of numerical solutions with respect to these trace formulas. Furthermore, we investigate in detail the influence of solution with different noise amplitude. The noise has an obvious effect on the phenomenon of phase shift and behavior of long time simulation of solution. Besides, the convergence order is considered with additive noise and multiplicative noise.

Section snippets

Preliminaries

In this section, we first provide some notations and lemmas used later in this work.

Lemma 2.1

cf. [12]

For uC[a,b], u(a)=u(b)=0, we make zero-extension such that u is defined on R. Suppose uC4+α(R){u|(1+|ξ|)4+α|uˆ(ξ)|dξ<},where uˆ(ξ) is the Fourier transform with respect to u(x), then for a fixed h, we have 43Δhαu(x)13Δ2hαu(x)=(Δ)α2u(x)+O(h4),Δhαu(x)=hαgkαu(xkh),1<α2,where gkα=(1)αΓ(α+1)Γ(α2k+1)Γ(α2+k+1),gkα=gkα,k=1,2.

Lemma 2.2

cf. [12]

There is a linear operator Λhα, such that (δhαu,v)=(Λhαu,Λhαv) for u,vVh.

Theorem 2.1

Construction of the fourth-order central difference scheme

For temporal and spatial discretizations, there exist two positive integers M and N such that h=baM and Δt=TN are space step and time size, respectively. The temporal–spatial domain D×[0,T] is covered by Ωh×ΩΔt={xj|xj=a+jh,j=0,2,,M}×{tk|tk=kΔt,k=0,1,,N}. Let Ujk={ujk|0jM,0kN,u0k=uMk=0} be grid function at the grid point (xj,tk). And some discrete inner products and corresponding norms are defined as follows (u,v)=i=1M1uivih,(δxu,δxv)=i=0M1(δxui+12)(δxvi+12)h,v=max0iM|vi|.Under

Numerical experiments

This section illustrates numerically the effect of noise on behavior of wave for the case of additive and multiplicative noise, we perform detailed numerical investigation based on the fourth-order central difference scheme. The evolution of energy in two kinds of noise is considered to verify the theoretical results. In following models, we consider F(u)=1cosu,u=u(x,t): Ω×[0,T]R with Ω=[30,30], T=30 the initial conditions are chosen as u(x,0), v(x,0)=ut(x,0), where u(x,t)=4arctan[1μ2cos(μ

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This project is supported by the National Natural Science Foundation of China (No. 11971241).

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