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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo


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Volume 16, Issue 1

Issues

Volume 13 (2015)

Stability and convergence of a local discontinuous Galerkin finite element method for the general Lax equation

Leilei Wei / Yundong Mu
Published Online: 2018-09-07 | DOI: https://doi.org/10.1515/math-2018-0091

Abstract

In this paper we develop and analyze the local discontinuous Galerkin (LDG) finite element method for solving the general Lax equation. The local discontinuous Galerkin method has the flexibility for arbitrary h and p adaptivity, and allows for hanging nodes. By choosing the numerical fluxes carefully we prove stability and give an error estimate. Finally some numerical examples are computed to show the convergence order and excellent numerical performance of proposed method.

Keywords: Lax equation; Local discontinuous Galerkin method; Stability analysis; Error estimates

MSC 2010: 35S10; 65M12

1 Introduction

In this paper we develop a local discontinuous Galerkin (LDG) method for general Lax equation

ut+αuuxxx+2αuxuxx+310α2u2ux+uxxxxx=0,u(x,0)=u0(x).(1)

where α are arbitrary nonzero and real parameters. We do not pay attention to boundary condition; hence the solution is considered to be either periodic or compactly supported.

There are only a few numerical works in the literature to solve the Lax equation. Xu and Shu [1] simulate the solutions of the Kawahara equation, the generalized Kawahara equation and Ito’s fifth-order mKdV equation. The general Lax equation discussed in our paper is different from the class of fifth-order equation in [1]. The general Lax equation (1) is an important mathematical model with wide applications in quantum mechanics, nonlinear optics, and describes motions of long waves in shallow water under gravity and in a one-dimensional nonlinear lattice [2,3,4]. Typical examples are widely used in various fields such as solid state physics, plasma physics, fluid physics and quantum field theory. It is well known that the Lax equation is completely integrable equation, and it has many sets of conservation laws [5,6]. For numerical study, Abdul-Majid Wazwaz [7] revealed solitons and periodic solutions for the fifth-order nonlinear KdV equation using the sine-cosine and the tanh methods. In [8] Cesar A studied the periodic and soliton solutions for the Lax equation using a generalization of extended tanh method. The DG method is beneficial for parallel computing and has high-order accuracy. Meanwhile, it is flexibility and efficiency in terms of mesh and shape functions. To our best knowledge, this is the first provably stable finite element method for the Lax equation.

The paper is organized as follows. In Section 2, notation and some preliminaries are described and cited. In Section 3, we discuss the LDG scheme for the general Lax equation, and prove the cell entropy inequality and L2 stability by choosing the fluxes carefully in Section 4. In Section 5 we give an error estimate, and some numerical experiments to illustrate the accuracy and capability of the method are given in Section 6. Concluding remarks are provided in Section 7.

2 Notations and auxiliary results

In this section we introduce notations and definitions to be used later in the paper and also present some auxiliary results.

2.1 Basic notations

We denote the mesh in [a, b] by Ij=(xj12,xj+12) for j = 1, 2⋯ N. The center of the cell is xj=12(xj12+xj+12) , and the mesh size is denoted hj=xj+12xj12 , with h = max1≤jN hj being the maximum mesh size. We assume that the mesh is regular, namely, that the ratio between the maximum and the minimum mesh sizes stays bounded during mesh refinements. We define the piecewise-polynomial space Vh as the space of polynomials of the degree up to k in each cell Ij, i.e.

Vh={v:vPk(Ij),xIj,j=1,2,N}.

Note that functions in Vh are allowed to have discontinuities across element interfaces.

The solution of the numerical scheme is denoted by uh, which belongs to the finite element space Vh. We denote by (uh)j+12+and(uh)j+12 the values of uh at xj+12 from the right cell Ij+1 and from the left cell Ij, respectively. [uh] is used to denote uh+uh , i.e. the jump of uh at cell interfaces.

Lemma 2.1

([9]). For any piecewise smooth function ωL2(Ω), on each cell boundary point we define

β(f^;ω)=β(f^;ω,ω+)=[ω]1(f(ω¯)f^(ω))if[ω]0,12|f(ω¯)|if[ω]=0,(2)

where (ω, ω+) is a monotone numerical flux consistent with the given flux f. Then β(;ω) is non-negative and bounded.

In the present paper we use C to denote a positive constant which may have a different value in each occurrence. The usual notation of norms in Sobolev spaces will be used. For any integer s ≥ 0, let Hs(Ω) represent the well-known Sobolev space equipped with the norm ∥⋅∥s, which consists of functions with (distributional) derivatives of order not greater than s in L2(Ω). Next, let the scalar inner product on L2 be denoted by (⋅, ⋅), and the associated norm by ∥⋅∥.

2.2 Projection

We will give the projection in one dimension [a, b], denoted by ℙ, i.e., for each j,

Ij(Pω(x)ω(x))v(x)=0,vPk(Ii),(3)

and special projection ℙ±, i.e., for each j,

Ij(P+ω(x)ω(x))v(x)=0,vPk1(Ij),andP+ω(xj12+)=ω(xj12),Ij(Pω(x)ω(x))v(x)=0,vPk1(Ij),andPω(xj+12)=ω(xj+12).(4)

There are some approximation results for the projection in [10,11,12]

ωe+hωe+h12ωeτhChk+1.(5)

where ωe = ℙωω or ωe = ℙ± ωω, and

ωeτh=(12i=1N[((ωe)+)i122+((ωe))i+122])12.

The positive constant C, solely depending on ω, is independent of h. τh denotes the set of boundary points of all elements Ij.

3 LDG Scheme

In this section, we define our LDG method for the general Lax equation (1), written in the following form:

ut+(α2u2)xxx(α2ux2)x+(α210u3)x+uxxxxx=0.(6)

To define the local discontinuous Galerkin method, we rewrite equation (1) as a first-order system:

f(u)=α210u3,B(z)=α2z2,b(z)=B(z)=αz,p=(b(z)u)x,zux=0,qzx=0,vqx=0,wvx=0,ut+f(u)x+pxB(z)x+wx=0.(7)

Now we can use the local discontinuous Galerkin method to equation (1), resulting in the following scheme: find uh, ph, wh, vh, qh, zhVh, such that for all test functions ρ, ϕ, φ, η, ξ, θVh

Ij(uh)tρdxIjf(uh)ρxdxIjphρxdx+IjB(zh)ρxdxIjwhρxdx+(f(uh)^ρ)j+12(f(uh)^ρ+)j12+(ph^ρ)j+12(ph^ρ+)j12(B(zh)^ρ)j+12+(B(zh)^ρ+)j12+(wh^ρ)j+12(wh^ρ+)j12=0,(8a)

Ijphϕdx+Ijb(zh)uhϕxdx(b(zh)^uh~ϕ)j+12+(b(zh)^uh~ϕ+)j12=0,(8b)

Ijwhφdx+Ijvhφxdx(vh^φ)j+12+(vh^φ+)j12=0,(8c)

Ijvhηdx+Ijqhηxdx(qh^η)j+12+(qh^η+)j12=0,(8d)

Ijqhξdx+Ijzhξxdx(zh^ξ)j+12+(zh^ξ+)j12=0,(8e)

Ijzhθdx+Ijuhθxdx(uh^θ)j+12+(uh^θ+)j12=0.(8f)

The “hat” terms in (8) in the cell boundary terms from integration by parts are the so-called “numerical fluxes”, which are single valued functions defined on the edges and should be designed based on different guiding principles for different PDEs to ensure stability. It turns out that we can take the simple choices such that

uh^=uh+,zh^=zh+τ1[vh],vh^=vhτ2[zh],ph^=ph,wh^=wh,qh^=qh,b(zh)^=B(zh+)B(zh)zh+zh,uh~=uh+,B(zh)^=B(zh),(9)

where τ1, τ2 > 0. Some dissipation terms in the flux of zh^,vh^ will give a control on the boundary terms. We have omitted the half-integer indices j12 as all quantities in (8) are computed at the same points (i.e. the interfaces between the cells). The flux is a monotone flux. Examples of monotone fluxes which are suitable for the local discontinuous Galerkin methods can be found in, e.g., [13]. For example, one could use the Lax-Friedriches flux, which is given by

f^LF(w,w+)=12(f(w)+f(w+)α(w+w)),α=maxw|f(w)|.(10)

We remark that the choice for the fluxes (9) is not unique. In fact the crucial part is taking uh^andph^,wh^ from opposite sides, taking vh^ except the dissipation term and zh^ except the dissipation term from opposite sides, taking B(zh)^anduh~ from opposite sides, and qh^=qh [14,15,16].

With such a choice of fluxes we can get the theoretical results of the L2 stability.

4 Stability analysis

Theorem 4.1

(cell entropy inequality). For periodic or compactly supported boundary conditions, the solution uh to the semi-discrete LDG scheme (8) satisfies the following cell entropy inequality

12ddtIjuh2dx+Φj+12Φj120.(11)

Proof

Choosing the test function θ = −wh in (8f), we obtain

IjzhwhdxIjuh(wh)xdx+(uh^wh)j12(uh^wh+)j12=0.(12)

Since (8) holds for any test functions in Vh, we can choose

ρ=uh,ϕ=zh,φ=zh,η=qh,ξ=vh,θ=ph(13)

Then we have

12ddtIjuh2dxIjf(uh)(uh)xdxIjph(uh)xdx+IjB(zh)(uh)xdxIjwh(uh)xdx+(f(uh)^uh)j+12(f(uh)^uh+)j12+(ph^uh)j+12(ph^uh+)j12(B(zh)^uh)j+12+(B(zh)^uh+)j12+(wh^uh)j+12(wh^uh+)j12=0,Ijphzhdx+Ijb(zh)uh(zh)xdx(b(zh)^uh~zh)j+12+(b(zh)^uh~zh+)j12=0,Ijwhzhdx+Ijvh(zh)xdx(vh^zh)j+12+(vh^zh+)j12=0,IjvhqhdxIjqh(qh)xdx+(qh^qh)j+12(qh^qh+)j12=0,Ijqhvhdx+Ijzh(vh)xdx(zh^vh)j+12+(zh^vh+)j12=0,IjzhphdxIjuh(ph)xdx+(uh^ph)j+12(uh^ph+)j12=0.(14)

Summing up Eqs. (12) and (14), we obtain

12ddtIjuh2dxF(uh)j+12+F(uh+)j12+(f(uh)^uh)j+12(f(uh)^uh+)j12+(B(zh)uh)j+12(B(zh+)u+)j12(B(zh)^uh)j+12+(B(zh)^uh+)j12(b(zh)^uh~zh)j+12+(b(zh)^uh~zh+)j12(phuh)j+12+(ph+uh+)j12+(ph^uh)j+12(ph^uh+)j12+(uh^ph)j+12(uh^ph+)j12(whuh)j+12+(wh+uh+)j12+(wh^uh)j+12(wh^uh+)j12+(uh^wh)j+12(uh^wh+)j12+(vhzh)j+12(vh+zh+)j12(vh^zh)j+12+(vh^zh+)j12(zh^vh)j+12+(zh^vh+)j1212(qh)j+122+12(qh+)j122+(qh^qh)j+12(qh^qh+)j12=0.(15)

where F(uh) = ∫uhf(s)ds. We introduce a short-hand notation

Φj+12=F(uh)j+12+(f(uh)^uh)j+12+(B(zh)u)j+12(B(zh)^u)j+12(b(zh)^uh~zh)j+12(phuh)j+12+(ph^uh)j+12+(uh^ph)j+12(whuh)j+12+(wh^uh)j+12+(uh^wh)j+12+(vhzh)j+12(vh^zh)j+12(zh^vh)j+1212(qh)j+122+(qh^qh)j+12(16)

Then we have

12ddtIjuh2dx+Φj+12Φj12+Θj12=0.(17)

and the extra term Θ is given by

Θj12=([F(uh)]f(uh)^[uh][B(zh)uh]+B(zh)^[uh]+b(zh)^uh~[zh]+[uhph]ph^[uh]uh^[ph]+[uhwh]wh^[uh]uh^[wh][vhzh]+vh^[zh]+zh^[vh]+[qh22]qh^[qh])j12+τ1[vh]2+τ2[zh]2.(18)

With the definition (9) of the numerical fluxes and after some algebraic manipulation, we easily obtain

[B(zh)uh]+B(zh)^[uh]+b(zh)^uh~[zh]=0[uhph]ph^[uh]uh^[ph]=0[uhwh]wh^[uh]uh^[wh]=0[vhzh]+vh^[zh]+zh^[vh]=0[qh22]qh^[qh]=12[qh]2

and hence

Θj12=([F(uh)]f(uh)^[uh])j120.(19)

where the last inequality follows from the monotonicity of the flux (10). This finishes the proof of the cell entropy inequality.  □

Summing up over Ij, we obtain the following L2 stability of numerical solution.

Theorem 4.2

(L2 stability). The solution u to the semi-discrete LDG scheme (8) satisfies the following L2 stability

12ddtabuh2dx0.

5 Error estimates

We state the main error estimates of the semi-discrete LDG scheme (8). We have the following theorem.

Theorem 5.1

Let u be the exact solution of the problem (1), which is sufficiently smooth with bounded derivatives. Let uh be the numerical solution of the semi-discrete LDG scheme (8). For rectangular triangulations of Ii × Jj, if the finite element space Vh is the piecewise polynomials of degree k ≥ 2, then for small enough h there holds the following error estimates

uuhChk.(20)

Proof

First we would like to make an a priori assumption that, for small enough h, there holds [17]

uuhh.(21)

Suppose that the interpolation property (5) is satisfied, then the a priori assumption (21) implies that

uuhCh12,QuuhCh12,(22)

where ℚ = ℙ or ℚ = ℙ± is the projection operator.

Notice that the equations (8) are also satisfied when the numerical solutions uh, ph, wh, vh, qh, zh are replaced by the exact solutions u, p, w, v, q, z. We then obtain the cell error equation

Ij(uuh)tρdxIj(f(u)f(uh))ρxdxIj(pph)ρxdx+Ij(B(z)B(zh))ρxdxIj(wwh)ρxdx+((f(u)f(uh)^)ρ)j+12((f(u)f(uh)^)ρ+)j12+((pph^)ρ)j+12((pph^)ρ+)j12((B(z)B(zh)^)ρ)j+12+((B(z)B(zh)^)ρ+)j12+((wwh^)ρ)j+12((wwh^)ρ+)j12+Ij(pph)ϕdx+Ij(b(z)ub(zh))uhϕxdx((b(z)ub(zh)^uh~)ϕ)j+12+((b(z)ub(zh)^uh~)ϕ+)j12+Ij(wwh)φdx+Ij(vvh)φxdx((vvh^)φ)j+12+((vvh^)φ+)j12+Ij(vvh)ηdx+Ij(qqh)ηxdx((qqh^)η)j+12+((qqh^)η+)j12+Ij(qqh)ξdx+Ij(zzh)ξxdx((zzh^)ξ)j+12+((zzh^)ξ+)j12+Ij(zzh)θdx+Ij(uuh)θxdx((uuh^)θ)j+12+((uuh^)θ+)j12=0.(23)

Define

Aj(uuh,pph,wwh,vvh,qqh,zzh;ρ,ϕ,φ,η,ξ,θ)=Ij(uuh)tρdxIj(pph)ρxdxIj(wwh)ρxdx+((pph^)ρ)j+12((pph^)ρ+)j12+((wwh^)ρ)j+12((wwh^)ρ+)j12+Ij(pph)ϕdx+Ij(wwh)φdx+Ij(vvh)φxdx((vvh^)φ)j+12+((vvh^)φ+)j12+Ij(vvh)ηdx+Ij(qqh)ηxdx((qqh^)η)j+12+((qqh^)η+)j12+Ij(qqh)ξdx+Ij(zzh)ξxdx((zzh^)ξ)j+12+((zzh^)ξ+)j12+Ij(zzh)θdx+Ij(uuh)θxdx((uuh^)θ)j+12+((uuh^)θ+)j12,(24)

Hj(f,u,uh;ρ)=Ij(f(u)f(uh))ρxdx((f(u)f(uh)^)ρ)j+12+((f(u)f(uh)^)ρ+)j12,(25)

and

Rj(B,b;z,u,zh,uh;ρ,ϕ)=Ij(B(z)B(zh))ρxdx+((B(z)B(zh)^)ρ)j+12((B(z)B(zh)^)ρ+)j12Ij(b(z)ub(zh))uhϕxdx+((b(z)ub(zh)^uh~)ϕ)j+12((b(z)ub(zh)^uh~)ϕ+)j12.(26)

Summing over j, the error equation (23) becomes

j=1NAij(uuh,pph,wwh,vvh,qqh,zzh;ρ,ϕ,φ,η,ξ,θ)=Hj(f,u,uh;ρ)+Rj(B,b;z,u,zh,uh;ρ,ϕ).(27)

Denoting

eu=uuh=P+uuh(P+uu)=P+eu(P+uu),ep=pph=Ppph(Ppp)=Pep(Ppp),ew=wwh=Pwwh(Pww)=Pew(Pww),ev=vvh=Pvvh(Pvv)=Pev(Pvv),eq=qqh=Pqqh(Pqq)=Peq(Pqq),ez=zzh=Pzzh(Pzz)=Pez(Pzz).

Take the test fuctions

ρ=P+eu,ϕ=Pez,φ=Pez,η=Peq,ξ=Pev,θ=(Pew+Pep),

from the linearity of 𝔄j we obtain the energy equality

j=1NAj(P+eu,Pep,Pew,Pev,Peq,Pez;P+eu,Pez,Pez,Peq,Pev,(Pew+Pep))=j=1NAj(P+uu,Ppp,Pww,Pvv,Pqq,Pzz;P+eu,Pez,Pez,Peq,Pev,(Pew+Pep))+Hj(f,u,uh;P+eu)+Rj(B,b;z,u,zh,uh;P+eu,Pez).(28)

First we consider the left-hand side of the energy equation (28).

The proof is by the same argument as that used for the stability result in Section 4.

The estimate for the second term of right-hand side in (28) is given in the following lemma.

For the proof of this lemma, we refer readers to Lemmas 3.4 and 3.5 in [18]. For the linear flux f(u) = cu, this a priori assumption is unnecessary, hence the result in Theorem 5.1 holds for any k ≥ 0.

The estimate for the final term of right-hand side in (28) is given in the following lemma.

For the proof of this lemma, we refer readers to Lemmas 4.6 in [17].

Plugging (29), (30), (34) and (35) into the equality (28), we can obtain

12ddtab(P+eu)2dx+j=1N(12[Peq]2+τ1[Pev]2+τ2[Pez]2)j12+j=1N14β(f^;uh)j12[P+eu]j122Cab12(P+eu)2dx+j=1Nε([P+eu]2+[Pez]2+[Peq]2+[Pev]2)j12+Ch2k+1+Ch2k,(36)

the fact that the initial error

u(,0)uh(,0)Chk+1,(37)

and the interpolating property (5) finally give us the error estimate (20).

To complete the proof, let us verify the a priori assumption (21). For k ≥ 1, we can consider h small enough so that Chk < 12h, where C is the constant in (20) determined by the final time T. Then, if t* = sup{t : ∥uuh∥ ≤ h}, we would have ∥u(t*) – uh(t*)∥ = h by continuity if t* is finite. On the other hand, our proof implies that (20) holds for tt*, in particular ∥u(t*) – uh(t*)∥ ≤ Chk < 12h. This is a contradiction if t* < T . Hence t*T and our a priori assumption (21) is justified. □

6 Numerical examples

In this section, we perform numerical experiments of the local discontinuous Galerkin method applied to the general Lax equation. We use the third order Runge-Kutta method and time steps are suitably adjusted in order to show a dominant spatial accuracy. All the computations were performed in double precision. This is not the most efficient method for the time discretization to our LDG scheme. However, we will not address the issue of time discretization efficiency in this paper. We have verified that the results shown are numerically convergent in all cases with the aid of successive mesh refinements.

Example 6.1

We consider the standard Lax equation (1) with α = 10 in I = [–5, 5], the exact solution is of the form

u(x,t)=1+3sech2(22(x14t))).(38)

The L2 and L1 errors and the numerical orders of accuracy at time t = 0.0001 are contained in Table 1. We can see that the method with Pk elements gives (k + 1)-th order of accuracy in both L2 and L1 norms.

Table 1

Accuracy test for Lax equation with the exact solution (38). Periodic boundary condition in [–5, 5]. Uniform meshes with N cells at final time T = 0.0001

Example 6.2

In this example, we test the scheme for the standard Lax equation with α = 10 in I = [–10, 10]. We take the soliton solutions of the form

u(x,t)=12csech2(12c4(xct)).(39)

We choose the constants c = 16. The L2 and L1 errors and the numerical orders of accuracy for u at time t = 0.0001 with uniform meshes are contained in Table 2. Periodic boundary conditions are used. We can see that the method with Pk elements gives a uniform (k + 1)-th order of accuracy for u in both norms.

Table 2

Accuracy test for Lax equation with the exact solution (39) choosing c = 16. Periodic boundary condition in [–10, 10]. Uniform meshes with N cells at time t = 0.0001

Example 6.3

In this example, we test the scheme for the Lax equation with α = 20. The solutions are of the form

u(x,t)=5λαsech2(λ42(x+λt)).(40)

We choose the constants λ = –16. The L2 and L1 errors and the numerical orders of accuracy for u at time t = 0.0001 with uniform meshes are contained in Table 3. Periodic boundary conditions are used. We can see that the method with Pk elements gives a uniform (k + 1)-th order of accuracy for u in both norms.

Table 3

Accuracy test for Lax equation with the exact solution (40) choosing λ = –16. Periodic boundary condition in [–5, 5]. Uniform meshes with N cells at time t = 0.0001

7 Conclusion

We have discussed the application of local discontinuous Galerkin methods to solve the general Lax equation. We prove stability and give an error estimate. Numerical examples for general Lax equation are given to illustrate the accuracy and capability of the methods. Although not addressed in this paper, the method is flexible for general geometry, unstructured meshes and h-p adaptivity, and has excellent parallel efficiency. These results indicate that the LDG method is a good tool for solving such nonlinear equations in mathematical physics.

Acknowledgement

This work is supported by the Foundation of Henan Educational Committee (19A110005), the Fundamental Research Funds for the Henan Provincial Colleges and Universities in Henan University of Technology (31490090), and the National Natural Science Foundation of China (11461072).

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About the article

leileiwei@haut.edu.cn


Received: 2016-10-08

Accepted: 2017-12-07

Published Online: 2018-09-07


Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 1091–1103, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0091.

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© 2018 Wei and Mu, published by De Gruyter.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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