A discontinuous Galerkin method and its error estimate for nonlinear fourth-order wave equations

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Abstract

In this paper, an ultra-weak local discontinuous Galerkin (UWLDG) method for a class of nonlinear fourth-order wave equations is designed and analyzed. The UWLDG method is a new DG method designed for solving partial differential equations (PDEs) with high order spatial derivatives. We prove the energy conserving property of our scheme and its optimal error estimates in the L2-norm for the solution itself as well as for the auxiliary variables approximating the derivatives of the solution. Compatible high order energy conserving time integrators are also proposed. The theoretical results are confirmed by numerical experiments.

Introduction

In recent years, many numerical methods have been defined and analyzed for the wave equations [1], [2], [3], [4], [5], [6], [7], [8], [9]. The nonlinear fourth-order wave equations arise commonly from the studies of vibration of beams and thin plates [10]. In this paper, we are interested in the numerical methods for a class of nonlinear fourth-order wave equations [11], [12], [13], [14], [15], [16], [17], [18], [19], utt+Δ2u+u+f(u)=0,xΩ,t[0,T],with the initial conditions u(x,0)=u0(x),ut(x,0)=v0(x).For the sake of simplicity, we only consider the periodic boundary condition. The solution u=u(x,t), xΩRd,d=1,2,t[0,T] is a real-valued function, and the initial conditions u0 and v0 are assumed to be as smooth as necessary. Levandosky [16] proved that our problem (1.1) admits a unique local solution for nonlinearity f(u) which satisfies f(0)=0;fC1(R)and|f(u)|c|u|p1,for 1<p21, where 2= for 1d4 and 2=2dd4 for d5, denotes the critical exponent for the embedding of H2(Rd) into Lq(Rd), 2q2.

There are many numerical methods proposed in the literature for solving the fourth-order equations [20], [21], [22], [23], [24], [25], [26]. In [20], Achouri designed a second-order conservative finite difference scheme for the two-dimensional fourth-order nonlinear wave equation. The mixed finite elements for the fourth-order wave equations also have been studied by He et al. in [25]. They considered mixed finite element method with explicit and implicit discretization in time and derived the optimal error estimate in the L2 norm. In [22], [23], Baccouch applied the local discontinuous Galerkin (LDG) method for the fourth-order Euler–Bernoulli partial differential equation (PDE) in one dimension, including superconvergence analysis and a posterior error estimate.

We consider an ultraweak-local discontinuous Galerkin (UWLDG) method introduced in [27] for (1.1). The DG method is a class of finite element methods using completely discontinuous basis functions. The first DG method was introduced in 1973 by Reed and Hill [28] in the framework of neutron transport. It was later developed for time-dependent nonlinear hyperbolic conservation laws, coupled with the Runge–Kutta time discretization, by Cockburn et al. [29], [30], [31]. Since then, the DG method has been intensively studied and successfully applied to various problems in a wide range of applications due to its flexibility with meshing, its compactness and its high parallel efficiency. The UWLDG method is a discontinuous Galerkin method designed for PDEs with high order spatial derivatives, which combines the advantage of the LDG method and the ultra-weak DG (UWDG) method. The idea of the LDG method [7], [32], [33], [34] is to rewrite the equations with higher order spatial derivatives into a first order system, then apply the DG method to this system and design suitable numerical fluxes to ensure stability. The UWDG method [35] is based on repeated integration by parts to move all spatial derivatives to the test function in the weak formulation, and to ensure stability by carefully choosing numerical fluxes. In our method, at first, we rewrite Eq. (1.1) as a second-order system. Then we repeat the application of integration by parts, and choose suitable numerical fluxes to ensure stability. Compared to the LDG method, we introduce fewer auxiliary variables, thereby reducing memory and computational costs. Compared to the UWDG method, we do not need any internal penalty terms to ensure stability.

We define the energy Eu=Ω12(ut)2+12(Δu)2+12u2+F(u)dx,where F(s)=f(s) and F(0)=0. For Eq. (1.1) Eu is a constant. Therefore, we would like to design a numerical method that conserves the energy Eu. Energy conserving DG methods for wave equations have been developed in [36], [37], [38], [39], [40]. Recently, Chou et al. [36], [40] developed an optimal energy-conserving local DG method for multi-dimensional second-order wave equation in heterogeneous media. Later, Fu and Shu [37] proposed an optimal energy conserving DG method for linear symmetric hyperbolic systems on general unstructured meshes. They proved a priori optimal error estimates for the semi-discrete scheme in one dimension, and also in multi-dimensions for Cartesian meshes when using tensor-product polynomials. They also proposed an energy-conserving ultra-weak DG method for the generalized Korteweg–de Vries (KdV) equations in one dimension [38], and proved its optimal error estimate. In this work, we design an optimally convergent energy-conserving method for the nonlinear fourth-order equations. We choose the alternating fluxes, and prove that the energy is conserved both in one-dimensional and two-dimensional cases. We also prove the optimal error estimates in the L2-norm for the solution itself as well as for the auxiliary variables.

The organization of the paper is as follows. In Section 2, we introduce some notations and the UWLDG method. In Section 3, the energy conserving property of our scheme will be discussed. In Section 4, we will introduce some projections and give the optimal error estimates in the L2-norm for one-dimensional and two-dimensional cases. Time discretization will be shown in Section 5. The theoretical results are confirmed numerically in Section 6. In Section 7, we give some concluding remarks.

Section snippets

Notations

Let us introduce some notations. Throughout this paper, we adopt standard notations for the Sobolev spaces such as Wm,q(D) on the subdomain DΩ equipped with the norm Wm,q(D). If D=Ω, we omit the index D; and if q=2, we set Wm,q(D)=Hm(D), Wm,q(D)=Hm(D); and we use D to denote the L2 norm in D.

Let Ωh denote a tessellation of Ω with shape-regular elements K, and the union of the boundary faces of elements KΩh, denoted as Ω=KΩhK. We denote the diameter of K by hK, and set h=maxKhK

Energy conservation

In this section, we will demonstrate that the UWLDG scheme (2.3)–(2.4) conserves the discrete energy. Experience shows that the scheme conserving the discrete energy can often behave better, especially in long time simulation.

Theorem 3.1

The energy Eh(t)=Ω12(uh)t2+12wh2+12uh2+F(uh)dx,is conserved by the semi-discrete UWLDG method (2.3)(2.4), with numerical fluxes (2.5)(2.8) for all time.

Proof

Without loss of generality, we choose the flux (2.5). In Eq. (2.3), we take the test function to be φ=(uh)t: ((uh)tt,(u

Error estimates

We study the optimal error estimates for the UWLDG method defined in (2.3)–(2.4) for Eq. (1.1). In Section 4.1, we introduce some projections and inequalities that will be used in our proof. In Section 4.2, we give the error estimate in the L2 norm.

Time discretization

In this section, we consider the fully discrete method of scheme (2.3)–(2.4). We use the UWLDG method for the spacial discretization, it can be of high order accuracy. Therefore, we also would like to introduce an explicit, energy conserving, high order time stepping method. As in LDG method, the auxiliary variable wh in our method could be solved in terms of uh in an element-by-element fashion. After eliminating wh, we can get a linear second-order ordinary differential system as follows: Müh(

Numerical examples

In this section, we present numerical examples to verify our theoretical convergence properties of the UWLDG method.

Example 6.1

First example, we consider the linear fourth-order equations in one-dimension with the periodic boundary condition. utt+uxxxx=0,(x,t)[0,2π]×(0,10],and the initial conditions u(x,0)=cos(x),ut(x,0)=sin(x).The exact solution of the problem is u(x,t)=cos(x+t).

We implement the DG method (2.3)–(2.4) with the alternating fluxes (2.5) and use the time discrete method (5.1). The errors

Concluding remarks

In this paper, we have developed an UWLDG method for a class of nonlinear fourth-order wave equations. The UWLDG methods combine the LDG and UWDG methods for solving time-dependent PDEs with high order spatial derivatives. The numerical fluxes have been carefully chosen to make our scheme energy conserving. We have proved the optimal error estimate in the L2-norm for the solution itself as well as for the auxiliary variables approximating its derivatives in the semi-discrete method, and have

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  • Cited by (6)

    1

    Research supported by China Scholarship Council.

    2

    Research supported by Science Challenge Project, China TZZT2019-A2.3, National Numerical Windtunnel, China Project NNW2019ZT4-B08, NSFC, China grants 11722112.

    3

    Research supported by NSF grant DMS-1719410.

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