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

  • Leilei Wei EMAIL logo und Yundong Mu
Veröffentlicht/Copyright: 7. September 2018
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Open Mathematics
Aus der Zeitschrift Open Mathematics Band 16 Heft 1

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=(xj−12,xj+12) for j = 1, 2⋯ N. The center of the cell is xj=12(xj−12+xj+12) , and the mesh size is denoted hj=xj+12−xj−12 , with h = max1≤j≤Nhj 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:v∈Pk(Ij),x∈Ij,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 functionω ∈ L2(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)

wheref̂(ω−, ω+) is a monotone numerical flux consistent with the given fluxf. Thenβ(f̂;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,∀v∈Pk(Ii),(3)

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

∫Ij(P+ω(x)−ω(x))v(x)=0,∀v∈Pk−1(Ij),and  P+ω(xj−12+)=ω(xj−12),∫Ij(P−ω(x)−ω(x))v(x)=0,∀v∈Pk−1(Ij),and  P−ω(xj+12−)=ω(xj+12).(4)

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

∥ωe∥+h∥ωe∥∞+h12∥ωe∥τh≤Chk+1.(5)

where ωe = ℙω − ω or ωe = ℙ±ω − ω, and

∥ωe∥τh=(12∑i=1N[((ωe)+)i−122+((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,z−ux=0,  q−zx=0,  v−qx=0,  w−vx=0,ut+f(u)x+px−B(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, zh ∈ Vh, such that for all test functions ρ, ϕ, φ, η, ξ, θ ∈ Vh

∫Ij(uh)tρdx−∫Ijf(uh)ρxdx−∫Ijphρxdx+∫IjB(zh)ρxdx−∫Ijwhρxdx+(f(uh)^ρ−)j+12−(f(uh)^ρ+)j−12+(ph^ρ−)j+12−(ph^ρ+)j−12−(B(zh)^ρ−)j+12+(B(zh)^ρ+)j−12+(wh^ρ−)j+12−(wh^ρ+)j−12=0,(8a)
∫Ijphϕdx+∫Ijb(zh)uhϕxdx−(b(zh)^uh~ϕ−)j+12+(b(zh)^uh~ϕ+)j−12=0,(8b)
∫Ijwhφdx+∫Ijvhφxdx−(vh^φ−)j+12+(vh^φ+)j−12=0,(8c)
∫Ijvhηdx+∫Ijqhηxdx−(qh^η−)j+12+(qh^η+)j−12=0,(8d)
∫Ijqhξdx+∫Ijzhξxdx−(zh^ξ−)j+12+(zh^ξ+)j−12=0,(8e)
∫Ijzhθdx+∫Ijuhθxdx−(uh^θ−)j+12+(uh^θ+)j−12=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^=vh−−τ2[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 j−12 as all quantities in (8) are computed at the same points (i.e. the interfaces between the cells). The flux f̂ 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^ and ph^, wh^ from opposite sides, taking vh^ except the dissipation term and zh^ except the dissipation term from opposite sides, taking B(zh)^ and uh~ 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 solutionuhto the semi-discrete LDG scheme (8) satisfies the following cell entropy inequality

12ddt∫Ijuh2dx+Φj+12−Φj−12≤0.(11)

Proof

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

−∫Ijzhwhdx−∫Ijuh(wh)xdx+(uh^wh−)j−12−(uh^wh+)j−12=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

  12ddt∫Ijuh2dx−∫Ijf(uh)(uh)xdx−∫Ijph(uh)xdx+∫IjB(zh)(uh)xdx −∫Ijwh(uh)xdx+(f(uh)^uh−)j+12−(f(uh)^uh+)j−12 +(ph^uh−)j+12−(ph^uh+)j−12−(B(zh)^uh−)j+12   +(B(zh)^uh+)j−12+(wh^uh−)j+12−(wh^uh+)j−12=0,  ∫Ijphzhdx+∫Ijb(zh)uh(zh)xdx−(b(zh)^uh~zh−)j+12+(b(zh)^uh~zh+)j−12=0,  ∫Ijwhzhdx+∫Ijvh(zh)xdx−(vh^zh−)j+12+(vh^zh+)j−12=0,−∫Ijvhqhdx−∫Ijqh(qh)xdx+(qh^qh−)j+12−(qh^qh+)j−12=0,∫Ijqhvhdx+∫Ijzh(vh)xdx−(zh^vh−)j+12+(zh^vh+)j−12=0,−∫Ijzhphdx−∫Ijuh(ph)xdx+(uh^ph−)j+12−(uh^ph+)j−12=0.(14)

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

12ddt∫Ijuh2dx−F(uh−)j+12+F(uh+)j−12+(f(uh)^uh−)j+12−(f(uh)^uh+)j−12+(B(zh−)uh−)j+12−(B(zh+)u+)j−12−(B(zh)^uh−)j+12+(B(zh)^uh+)j−12−(b(zh)^uh~zh−)j+12+(b(zh)^uh~zh+)j−12−(ph−uh−)j+12+(ph+uh+)j−12+(ph^uh−)j+12−(ph^uh+)j−12+(uh^ph−)j+12−(uh^ph+)j−12−(wh−uh−)j+12+(wh+uh+)j−12+(wh^uh−)j+12−(wh^uh+)j−12+(uh^wh−)j+12−(uh^wh+)j−12+(vh−zh−)j+12−(vh+zh+)j−12−(vh^zh−)j+12+(vh^zh+)j−12−(zh^vh−)j+12+(zh^vh+)j−12−12(qh−)j+122+12(qh+)j−122+(qh^qh−)j+12−(qh^qh+)j−12=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−(ph−uh−)j+12+(ph^uh−)j+12+(uh^ph−)j+12−(wh−uh−)j+12+(wh^uh−)j+12+(uh^wh−)j+12+(vh−zh−)j+12−(vh^zh−)j+12−(zh^vh−)j+12−12(qh−)j+122+(qh^qh−)j+12(16)

Then we have

12ddt∫Ijuh2dx+Φj+12−Φj−12+Θj−12=0.(17)

and the extra term Θ is given by

Θj−12=([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])j−12+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

Θj−12=([F(uh)]−f(uh)^[uh])j−12≥0.(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 solutionuto the semi-discrete LDG scheme (8) satisfies the followingL2stability

12ddt∫abuh2dx≤0.

5 Error estimates

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

Theorem 5.1

Letube the exact solution of the problem (1), which is sufficiently smooth with bounded derivatives. Letuhbe the numerical solution of the semi-discrete LDG scheme (8). For rectangular triangulations of Ii × Jj, if the finite element spaceVhis the piecewise polynomials of degree k ≥ 2, then for small enoughhthere holds the following error estimates

∥u−uh∥≤Chk.(20)

Proof

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

∥u−uh∥≤h.(21)

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

∥u−uh∥∞≤Ch12,   ∥Qu−uh∥∞≤Ch12,(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(u−uh)tρdx−∫Ij(f(u)−f(uh))ρxdx−∫Ij(p−ph)ρxdx+∫Ij(B(z)−B(zh))ρxdx−∫Ij(w−wh)ρxdx+((f(u)−f(uh)^)ρ−)j+12−((f(u)−f(uh)^)ρ+)j−12+((p−ph^)ρ−)j+12−((p−ph^)ρ+)j−12−((B(z)−B(zh)^)ρ−)j+12+((B(z)−B(zh)^)ρ+)j−12+((w−wh^)ρ−)j+12−((w−wh^)ρ+)j−12+∫Ij(p−ph)ϕdx+∫Ij(b(z)u−b(zh))uhϕxdx−((b(z)u−b(zh)^uh~)ϕ−)j+12+((b(z)u−b(zh)^uh~)ϕ+)j−12+∫Ij(w−wh)φdx+∫Ij(v−vh)φxdx−((v−vh^)φ−)j+12+((v−vh^)φ+)j−12+∫Ij(v−vh)ηdx+∫Ij(q−qh)ηxdx−((q−qh^)η−)j+12+((q−qh^)η+)j−12+∫Ij(q−qh)ξdx+∫Ij(z−zh)ξxdx−((z−zh^)ξ−)j+12+((z−zh^)ξ+)j−12+∫Ij(z−zh)θdx+∫Ij(u−uh)θxdx−((u−uh^)θ−)j+12+((u−uh^)θ+)j−12=0.(23)

Define

Aj(u−uh,p−ph,w−wh,v−vh,q−qh,z−zh;ρ,ϕ,φ,η,ξ,θ)=∫Ij(u−uh)tρdx−∫Ij(p−ph)ρxdx−∫Ij(w−wh)ρxdx+((p−ph^)ρ−)j+12−((p−ph^)ρ+)j−12+((w−wh^)ρ−)j+12−((w−wh^)ρ+)j−12+∫Ij(p−ph)ϕdx+∫Ij(w−wh)φdx+∫Ij(v−vh)φxdx−((v−vh^)φ−)j+12+((v−vh^)φ+)j−12+∫Ij(v−vh)ηdx+∫Ij(q−qh)ηxdx−((q−qh^)η−)j+12+((q−qh^)η+)j−12+∫Ij(q−qh)ξdx+∫Ij(z−zh)ξxdx−((z−zh^)ξ−)j+12+((z−zh^)ξ+)j−12+∫Ij(z−zh)θdx+∫Ij(u−uh)θxdx−((u−uh^)θ−)j+12+((u−uh^)θ+)j−12,(24)
Hj(f,u,uh;ρ)=∫Ij(f(u)−f(uh))ρxdx−((f(u)−f(uh)^)ρ−)j+12+((f(u)−f(uh)^)ρ+)j−12,(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+)j−12 −∫Ij(b(z)u−b(zh))uhϕxdx+((b(z)u−b(zh)^uh~)ϕ−)j+12−((b(z)u−b(zh)^uh~)ϕ+)j−12.(26)

Summing over j, the error equation (23) becomes

∑j=1NAij(u−uh,p−ph,w−wh,v−vh,q−qh,z−zh;ρ,ϕ,φ,η,ξ,θ)  =Hj(f,u,uh;ρ)+Rj(B,b;z,u,zh,uh;ρ,ϕ).(27)

Denoting

eu=u−uh=P+u−uh−(P+u−u)=P+eu−(P+u−u),ep=p−ph=Pp−ph−(Pp−p)=Pep−(Pp−p),ew=w−wh=Pw−wh−(Pw−w)=Pew−(Pw−w),  ev=v−vh=Pv−vh−(Pv−v)=Pev−(Pv−v),eq=q−qh=Pq−qh−(Pq−q)=Peq−(Pq−q),  ez=z−zh=Pz−zh−(Pz−z)=Pez−(Pz−z).

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+u−u,Pp−p,Pw−w,Pv−v,Pq−q,Pz−z;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).

Lemma 5.2

The following equation holds

∑j=1NAj(P+eu,Pep,Pew,Pev,Peq,Pez;P+eu,Pez,Pez,−Peq,Pev,−(Pew+Pep)) =12ddt∫ab(P+eu)2dx+∑j=1N(12[Peq]2+τ1[Pev]2+τ2[Pez]2)j−12.(29)

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

Lemma 5.3

There exist numerical entropy fluxesBj+12,such that the following equation holds

Aj(P+u−u,Pp−p,Pw−w,Pv−v,Pq−q,Pz−z;P+eu,Pez,Pez,−Peq,Pev,−(Pew+Pep))≤∫Ij12(P+eu)2dx+Bj+12+Bj−12+ε([P+eu]2+[Pez]2+[Peq]2+[Pev]2)j−12.(30)

Proof

As to the first term of right-hand side in (28), we first write out all the terms

Aj(P+u−u,Pp−p,Pw−w,Pv−v,Pq−q,Pz−z;P+eu,Pez,Pez,−Peq,Pev,−(Pew+Pep))=∫Ij(P+u−u)tP+eudx−∫Ij(Pp−p)(P+eu)xdx−∫Ij(Pw−w)(P+eu)xdx+((Pp−p)−(P+eu)−)j+12−((Pp−p)−(P+eu)+)j−12+((Pw−w)−(P+eu)−)j+12−((Pw−w)−(P+eu)+)j−12+∫Ij(Pp−p)Pezdx+∫Ij(Pw−w)Pezdx+∫Ij(Pv−v)(Pez)xdx−((Pv−v)−(Pez)−)j+12+((Pv−v)−(Pez)+)j−12+∫Ij(Pv−v)(−Peq)dx+∫Ij(Pq−q)(−Peq)xdx−((Pq−q)−(−Peq)−)j+12+((Pq−q)−(−Peq)+)j−12+∫Ij(Pq−q)Pevdx+∫Ij(Pz−z)(Pev)xdx−((Pz−z)+(Pev)−)j+12+((Pz−z)+(Pev)+)j−12+∫Ij(z−zh)(−(Pew+Pep)))dx+∫Ij(P+u−u)(−(Pew+Pep)))xdx−((P+u−u)+(−(Pew+Pep)))−)j+12+((P+u−u)+(−(Pew+Pep)))+)j−12.(31)

Using the property of the special projection ℙ, ℙ– and ℙ+, the expression (31) becomes

Aj(P+u−u,Pp−p,Pw−w,Pv−v,Pq−q,Pz−z;P+eu,Pez,Pez,−Peq,Pev,−(Pew+Pep))=∫Ij(P+u−u)tP+eudx+((Pp−p)−(P+eu)−)j+12−((Pp−p)−(P+eu)+)j−12+((Pw−w)−(P+eu)−)j+12−((Pw−w)−(P+eu)+)j−12
−((Pv−v)−(Pez)−)j+12+((Pv−v)−(Pez)+)j−12−((Pq−q)−(−Peq)−)j+12+((Pq−q)−(−Peq)+)j−12−((Pz−z)+(Pev)−)j+12+((Pz−z)+(Pev)+)j−12≤∫Ij12(P+eu)2dx+Bj+12+Bj−12+ε([P+eu]2+[Pez]2+[Peq]2+[Pev]2)j−12(32)

where

B=(Pp−p)−(P+eu)−+(Pw−w)−(P+eu)−−(Pv−v)−(Pez)−−(Pq−q)−(−Peq)−−(Pz−z)+(Pev)−.(33)

Then we finish the proof of Lemma 5.3. □

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

Lemma 5.4

([18]). Suppose that the interpolation property (5) is satisfied; then we have the following estimate for

∑j=1NHj(f;u,uh;P+eu)≤∑j=1N−14β(f^;uh)j−12[P+eu]j−122+C∫ab(P+eu)2dx+Ch2k+1.(34)

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.

Lemma 5.5

([17]). Suppose that the interpolation property (5) is satisfied; then we have the following estimate for

|∑j=1NRj(B,b;z,u,zh,uh;P+eu,Pez)|≤∑j=1N(|b′(z)u(Pz−z)[Pez]|+|b(z)(Pz−z)−[P+eu]|)j−12+12∥P+eu∥2+C∥Pez∥2+Ch2k+2.(35)

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

12ddt∫ab(P+eu)2dx+∑j=1N(12[Peq]2+τ1[Pev]2+τ2[Pez]2)j−12+∑j=1N14β(f^;uh)j−12[P+eu]j−122≤C∫ab12(P+eu)2dx+∑j=1Nε([P+eu]2+[Pez]2+[Peq]2+[Pev]2)j−12+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 : ∥u–uh∥ ≤ 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 t ≤ t*, 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+3 sech2(22(x−14t))).(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

NL2-errororderL1-errororder
P0300.250025095519879-0.505252806535136-
350.2144568851908371.000.4305767209474501.04
400.1877349146418321.000.3785413787300340.96
450.1669274356926661.000.3353443385611881.03
500.1502681687089051.000.3026867283898710.97
P1301.879039441453699E-002-3.286545740223012E-002-
351.517793447466294E-0021.392.648611321788866E-0021.40
401.175185014958489E-0021.922.051650636107464E-0021.91
459.301100063281460E-0031.991.623421704551745E-0021.99
507.543588974233264E-0031.991.316444163568550E-0021.99
P2305.437341211792110E-003-9.452351537362624E-003-
352.586643947316112E-0034.824.388740273521469E-0034.98
401.634177482372270E-0033.442.801872130043346E-0033.36
451.131356506174137E-0033.121.911467182749563E-0033.25
508.272791940860751E-0042.971.442019800374107E-0032.67

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)=12c sech2(12c4(x−ct)).(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

NL2-errororderL1-errororder
P0450.262562836987084-0.442363863105017-
500.2367251148627930.980.4074355754922290.78
550.2154850370227660.990.3639858471573291.18
600.1977245275835030.990.3387928049200910.82
650.1826565428184170.990.3089650963381551.15
700.1697142250084660.990.2900197310475810.85
P1453.880589332295393E-002-5.917128300448593E-002-
503.206033525121685E-0021.814.853508770943189E-0021.88
552.671598323277074E-0021.913.977612736077588E-0022.09
602.262802545703084E-0021.913.386667345536551E-0021.85
651.940528673241487E-0021.922.904070662670210E-0021.92
701.682010624218387E-0021.932.485815847211677E-0022.10
P2451.506187123768599E-002-2.442327109945212E-002-
501.019076038429557E-0023.711.683162047008102E-0023.53
556.815580704424561E-0034.221.089819034720809E-0024.56
604.952438129993914E-0033.677.619139588922541E-0034.11
653.872358100972614E-0033.075.450917599415668E-0034.18
703.191727848648557E-0032.614.358328765569666E-0033.02

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)=5−λαsech2(−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

NL2-errororderL1-errororder
P0309.885941626628193E-002-0.169366079037095-
358.485448377283446E-0020.990.1435012087688241.08
407.431484018739452E-0020.990.1267556847460740.93
456.609867111817130E-0020.990.1119156295671311.06
505.951526640818351E-0021.000.1013065030825840.95
P1301.019279381151275E-002-1.535758890466971E-002-
358.340246396509980E-0031.301.239027721246629E-0021.39
406.482260809629124E-0031.899.583787194704823E-0031.92
455.146246954540782E-0031.967.574137237557694E-0032.00
504.183737195964606E-0031.976.140217793902676E-0032.00
P2302.704100139391892E-003-3.924975265062904E-003-
351.514846548183961E-0033.762.201195114825479E-0033.75
408.785309061922388E-0044.081.282439894082946E-0034.05
455.986846947394631E-0043.267.895185077491700E-0044.11
504.319620667412239E-0043.105.491810801293127E-0043.45

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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Received: 2016-10-08
Accepted: 2017-12-07
Published Online: 2018-09-07

© 2018 Wei and Mu, published by De Gruyter.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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https://doi.org/10.1515/math-2018-0091
Schlagwörter für diesen Artikel
Lax equation; Local discontinuous Galerkin method; Stability analysis; Error estimates
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Artikel in diesem Heft

  1. Regular Articles
  2. Algebraic proofs for shallow water bi–Hamiltonian systems for three cocycle of the semi-direct product of Kac–Moody and Virasoro Lie algebras
  3. On a viscous two-fluid channel flow including evaporation
  4. Generation of pseudo-random numbers with the use of inverse chaotic transformation
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  75. Nontrivial periodic solutions to delay difference equations via Morse theory
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  89. Stability and convergence of a local discontinuous Galerkin finite element method for the general Lax equation
  90. New topology in residuated lattices
  91. Optimality and duality in set-valued optimization utilizing limit sets
  92. An improved Schwarz Lemma at the boundary
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  95. Epi-mild normality
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  106. Uniqueness theorems for L-functions in the extended Selberg class
  107. An effective algorithm for globally solving quadratic programs using parametric linearization technique
  108. Bounds of Strong EMT Strength for certain Subdivision of Star and Bistar
  109. On categorical aspects of S -quantales
  110. On the algebraicity of coefficients of half-integral weight mock modular forms
  111. Dunkl analogue of Szász-mirakjan operators of blending type
  112. Majorization, “useful” Csiszár divergence and “useful” Zipf-Mandelbrot law
  113. Global stability of a distributed delayed viral model with general incidence rate
  114. Analyzing a generalized pest-natural enemy model with nonlinear impulsive control
  115. Boundary value problems of a discrete generalized beam equation via variational methods
  116. Common fixed point theorem of six self-mappings in Menger spaces using (CLRST) property
  117. Periodic and subharmonic solutions for a 2nth-order p-Laplacian difference equation containing both advances and retardations
  118. Spectrum of free-form Sudoku graphs
  119. Regularity of fuzzy convergence spaces
  120. The well-posedness of solution to a compressible non-Newtonian fluid with self-gravitational potential
  121. On further refinements for Young inequalities
  122. Pretty good state transfer on 1-sum of star graphs
  123. On a conjecture about generalized Q-recurrence
  124. Univariate approximating schemes and their non-tensor product generalization
  125. Multi-term fractional differential equations with nonlocal boundary conditions
  126. Homoclinic and heteroclinic solutions to a hepatitis C evolution model
  127. Regularity of one-sided multilinear fractional maximal functions
  128. Galois connections between sets of paths and closure operators in simple graphs
  129. KGSA: A Gravitational Search Algorithm for Multimodal Optimization based on K-Means Niching Technique and a Novel Elitism Strategy
  130. θ-type Calderón-Zygmund Operators and Commutators in Variable Exponents Herz space
  131. An integral that counts the zeros of a function
  132. On rough sets induced by fuzzy relations approach in semigroups
  133. Computational uncertainty quantification for random non-autonomous second order linear differential equations via adapted gPC: a comparative case study with random Fröbenius method and Monte Carlo simulation
  134. The fourth order strongly noncanonical operators
  135. Topical Issue on Cyber-security Mathematics
  136. Review of Cryptographic Schemes applied to Remote Electronic Voting systems: remaining challenges and the upcoming post-quantum paradigm
  137. Linearity in decimation-based generators: an improved cryptanalysis on the shrinking generator
  138. On dynamic network security: A random decentering algorithm on graphs
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Heruntergeladen am 7.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/math-2018-0091/html
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