Open access peer-reviewed chapter

Local Discontinuous Galerkin Method for Nonlinear Ginzburg- Landau Equation

By Tarek Aboelenen

Submitted: November 18th 2017Reviewed: February 12th 2018Published: May 23rd 2018

DOI: 10.5772/intechopen.75300

Downloaded: 290

Abstract

The Ginzburg-Landau equation has been applied widely in many fields. It describes the amplitude evolution of instability waves in a large variety of dissipative systems in fluid mechanics, which are close to criticality. In this chapter, we develop a local discontinuous Galerkin method to solve the nonlinear Ginzburg-Landau equation. The nonlinear Ginzburg-Landau problem has been expressed as a system of low-order differential equations. Moreover, we prove stability and optimal order of convergence OhN+1 for Ginzburg-Landau equation where h and N are the space step size and polynomial degree, respectively. The numerical experiments confirm the theoretical results of the method.

Keywords

  • Ginzburg-Landau equation
  • discontinuous Galerkin method
  • stability
  • error estimates

1. Introduction

The Ginzburg-Landau equation has arisen as a suitable model in physics community, which describes a vast variety of phenomena from nonlinear waves to second-order phase transitions, from superconductivity, superfluidity, and Bose-Einstein condensation to liquid crystals and strings in field theory [1]. The Taylor-Couette flow, Bénard convection [1] and plane Poiseuille flow [2] are such examples where the Ginzburg-Landau equation is derived as a wave envelop or amplitude equation governing wave-packet solutions. In this chapter, we develop a nodal discontinuous Galerkin method to solve the nonlinear Ginzburg-Landau equation

utν+Δu+κ+u2uγu=0,E1

and periodic boundary conditions and η,ζ,γare real constants, ν,κ>0. Notice that the assumption of periodic boundary conditions is for simplicity only and is not essential: the method as well as the analysis can be easily adapted for nonperiodic boundary conditions.

The various kinds of numerical methods can be found for simulating solutions of the nonlinear Ginzburg-Landau problems [3, 4, 5, 6, 7, 8, 9, 10, 11]. The local discontinuous Galerkin (LDG) method is famous for high accuracy properties and extreme flexibility [12, 13, 14, 15, 16, 17, 18, 19, 20]. To the best of our knowledge, however, the LDG method, which is an important approach to solve partial differential equations, has not been considered for the nonlinear Ginzburg-Landau equation. Compared with finite difference methods, it has the advantage of greatly facilitating the handling of complicated geometries and elements of various shapes and types as well as the treatment of boundary conditions. The higher order of convergence can be achieved without many iterations.

The outline of this chapter is as follows. In Section 2, we derive the discontinuous Galerkin formulation for the nonlinear Ginzburg-Landau equation. In Section 3, we prove a theoretical result of L2 stability for the nonlinear case as well as an error estimate for the linear case. Section 4 presents some numerical examples to illustrate the efficiency of the scheme. A few concluding remarks are given in Section 5.

2. LDG scheme for Ginzburg-Landau equation

In order to construct the LDG method, we rewrite the second derivative as first-order derivatives to recover the equation to a low-order system. However, for the first-order system, central fluxes are used. We introduce variables r,sand set

r=xs,s=xu,E2

then, the Ginzburg-Landau problem can be rewritten as

utν+r+κ+u2uγu=0,r=xs,s=xu.E3

We consider problem posed on the physical domain Ωwith boundary ∂Ωand assume that a nonoverlapping element Dksuch that

Ω=k=1KDk.E4

Now we introduce the broken Sobolev space for any real number r

HrΩ={vL2Ω:k=12.KvDkHrDk}.E5

We define the local inner product and L2Dknorm

uvDk=Dkuvdx,uDk2=uuDk,E6

as well as the global broken inner product and norm

uvΩ=k=1KuvDk,uL2Ω2=k=1KuuDk.E7

We define the jumps along a normal, n̂, as

u=n̂u+n̂+u+.E8

The numerical traces (u,s) are defined on interelement faces as the central fluxes

u=u=u++u2,s=s=s++s2.E9

Let us discretize the computational domain Ωinto Knonoverlapping elements, Dk=xk12xk+12, Δxk=xk+12xk12and k=1,,K. We assume uh,rh,shVkNbe the approximation of u,r,srespectively, where the approximation space is defined as

VkN=v:vkPNDkDkΩ,E10

where PNDkdenotes the set of polynomials of degree up to N defined on the element Dk. We define local discontinuous Galerkin scheme as follows: find uh,rh,shVkN, such that for all test functions ϑ,ϕ,φVkN,

uhtϑDkν+rhϑDk+κ+uh2uhϑDkγuhϑDk=0,rhϕDk=xshϕDk,shφDk=xuhφDk.E11

Applying integration by parts to (11), and replacing the fluxes at the interfaces by the corresponding numerical fluxes, we obtain

uhtϑDkν+rhϑDk+κ+uh2uhϑDkγuhϑDk=0,rhϕDk=shϕxDk+shϕk+12shϕ+k12,shφDk=uhφxDk+uhφk+12uhφ+k12,E12

we can rewrite (12) as

uhtϑDkν+rhϑDk+κ+uh2uhϑDkγuhϑDk=0,rhϕDk=shϕxDk+n̂.shϕDk,shφDk=uhφxDk+n̂.uhφDk.E13

where n̂is simply a scalar and takes the value of +1 and −1 at the right and the left interface, respectively.

3. Stability and error estimates

In this section, we discuss stability and accuracy of the proposed scheme, for the Ginzburg-Landau problem.

3.1. Stability analysis

In order to carry out the analysis of the LDG scheme, we have the following results.

Theorem 3.1. (L2 stability). The solution given by the LDG method defined by (13) satisfies

uhxTΩe2γTu0xΩ

for any T>0.

Proof. Set ϑϕφ=uhνuhνshin (13) and consider the integration by parts formula (u,rx)Dk+(r,ux)Dk=[ur]xk12xk+12, we get

uhtuhDk+shshDk=νrhuhDk+ν+rhuhDkκ+uh2uhuhDk+γuhuhDk+νn̂.shuhDk+νn̂.uhshDkνn̂.shuhDk.E14

Taking the real part of the resulting equation, we obtain

uhtuhDk+shshDk=κuh2uhuhDk+γuhuhDk+νn̂.shuhDk+νn̂.uhshDkνn̂.shuhDk.E15

Removing the positive term κuh2uhuhDk, we obtain

uhtuhDk+shshDkγuhL2Dk2+νn̂.shuhDk+νn̂.uhshDkνn̂.shuhDk.E16

Summing over all elements (16), we easily obtain

uhtuhL2Ω+shshL2ΩγuhΩ2.E17

Employing Gronwall’s inequality, we obtain

uhxTΩe2γTu0xΩ.

3.2. Error estimates

We consider the linear Ginzburg-Landau equation

utν+Δu+κ+uγu=0.E18

It is easy to verify that the exact solution of the above (18) satisfies

utϑDkν+rϑDk+κ+uϑDkγuϑDk=0,rϕDk=sϕxDk+n̂.sϕDk,sφDk=uφxDk+n̂.uφDk.E19

Subtracting (19) from the linear Ginzburg-Landau Eq. (13), we have the following error equation

uuhtϑDk+sshϕxDk+uuhφxDk+κ+uuhϑDkγuuhϑDk+rrhϕDk+sshφDkn̂.sshϕDkν+rrhϑDkn̂.uuhφDk=0.E20

For the error estimate, we define special projections Pand P+into Vhk. For all the elements, Dk, k=1,2,,Kare defined to satisfy

P+uuvDk=0,vPNkDk,P+uxk12=uxk12,PuuvDk=0,vPNk1Dk,Puxk+12=uxk+12.E21

Denoting

π=Puuh,πe=Puu,ε=P+rrh,εe=P+rr,τ=P+ssh,τe=P+ss.E22

For the abovementioned special projections, we have, by the standard approximation theory [21], that

P+u.u.L2ΩhChN+1,Pu.u.L2ΩhChN+1,E23

where here and below C is a positive constant (which may have a different value in each occurrence) depending solely on u and its derivatives but not of h.

Theorem 3.2. Let u be the exact solution of the problem (18), and let uhbe the numerical solution of the semi-discrete LDG scheme (13). Then for small enough h, we have the following error estimates:

u.tuh.tL2ΩhChN+1,E24

where the constant C is dependent upon T and some norms of the solutions.

Proof. From the Galerkin orthogonality (20), we get

ππetϑDk+ττeϕxDk+ππeφxDk+κ+ππeϑDkγππeϑDk+εεeϕDk+ττeφDk+ϕϕeβDkn̂.ττeϕDkν+εεeϑDkn̂.ππeφDk=0.E25

Taking the real part of the resulting equation, we obtain

ππetϑDk+ττeϕxDk+ππeφxDk+κππeϑDkγππeϑDk+εεeϕDk+ττeφDkn̂.ττeϕDkνεεeϑDkn̂.ππeφDk=0.E26

We take the test functions

ϑ=π,ϕ=νπ,φ=ντ,E27

we obtain

ππetπDk+νττeπxDk+νππeτxDk+κππeπDkγππeπDk+νεεeπDk+νττeτDkνn̂.ττeπDkνεεeπDkνn̂.ππeτDk=0.E28

Summing over k, simplify by integration by parts and (9), we get

πtπΩ+νττΩ=ντeπxΩ+νπeτxΩ+πteπΩγπeπΩ+κπeπΩ+ντeτΩ+γππΩκππΩνk=1Kn̂.πeτDkνk=1Kn̂.τeπDk,E29

we can rewrite (29) as

πtπΩ+νττΩ=I+II+III,E30

where

I=ντeπxΩ+νπeτxΩ,E31
II=πteπΩγπeπΩ+κπeπΩ+ντeτΩνk=1Kn̂.πeτDkνk=1Kn̂.τeπDk,E32
III=γππΩκππΩ.E33

Using the definitions of the projections P,S(21) in (31), we get

I=0.E34

From the approximation results (23) and Young’s inequality in (32), we obtain

IIc1πL2Ω2+c2τL2Ω2+Ch2N+2.E35

and

IIIc1πL2Ω2.E36

Combining (34), (35), (36) and (30), we obtain

πtπΩ+νττΩc1πL2Ω2+c2τL2Ω2+Ch2N+2,E37

provided c2 is sufficiently small such that c2ν, we obtain that

πtπΩc1πL2Ω2+Ch2N+2.E38

From the Gronwall’s lemma and standard approximation theory, the desired result follows. ⃞.

4. Numerical examples

In this section, we present several numerical examples to illustrate the previous theoretical results. We use the high-order Runge-Kutta time discretizations [22], when the polynomials are of degree N, a higher order accurate Runge-Kutta (RK) method must be used in order to guarantee that the scheme is stable. In this chapter, we use a fourth-order non-total variation diminishing (TVD) Runge-Kutta scheme [23]. Numerical experiments demonstrate its numerical stability

uht=Fuht,E39

where uhis the vector of unknowns, we can use the standard fourth-order four-stage explicit RK method (ERK)

k1=Fuhntn,k2=Fuhn+12Δtk1tn+12Δt,k3=Fuhn+12Δtk2tn+12Δt,k4=Fuhn+Δtk3tn+Δt,uhn+1=uhn+16k1+2k2+2k3+k4,E40

to advance from uhnto uhn+1, separated by the time step, Δt. In our examples, the condition ΔtCΔxminα0<C<1is used to ensure stability.

Example 4.1 We consider the following linear Ginzburg-Landau equation

utν+Δu+κ+u=0,x2020,t0,0.5,ux0=u0x,E41

with

η=12,κ=ν31+4ν2122+9ν2,ζ=1,γ=0.E42

The exact solution uxt=axeidlnaxiωtwhere

ax=Fsechx,F=d1+4ν22κ,d=1+4ν212ν,ω=d1+4ν22ν.E43

The convergence rates and the numerical L2 error are listed in Figure 1 for several different values of ν, confirming optimal OhN+1order of convergence across.

Example 4.2 We consider the nonlinear Ginzburg-Landau Eq. (1) with initial condition,

ux0=ex2,E44

with parameters ν=1,κ=1,η=1,ζ=2, x1010. We consider cases with N = 2 and K = 40 and solve the equation for several different values of γ. The numerical solution uhxtfor γ=2,1,0,1,2is shown in Figures 2 and 3. The parameter γwill affect the wave shape. From these figures, it is obvious that the solution decays rapidly with time evolution especially for γ<0and the parameter γdramatically affects the wave shape.

Figure 1.

The rate of convergence for the solving the nonlinear Ginzburg-Landau equation in Example 4.2.

Figure 2.

Numerical results for the nonlinear Ginzburg-Landau equation in Example 4.2.

Figure 3.

Numerical results for the nonlinear Ginzburg-Landau equation with γ = −2 in Example 4.2.

5. Conclusions

In this chapter, we developed and analyzed a local discontinuous Galerkin method for solving the nonlinear Ginzburg-Landau equation and have proven the stability of this method. Numerical experiments confirm that the optimal order of convergence is recovered. As a last example, the Ginzburg-Landau equation with initial condition is solved for different values of γ and results show that the parameter γ dramatically affects the wave shape. In addition, the solution decays rapidly with time evolution especially for γ<0.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Tarek Aboelenen (May 23rd 2018). Local Discontinuous Galerkin Method for Nonlinear Ginzburg- Landau Equation, Differential Equations - Theory and Current Research, Terry E. Moschandreou, IntechOpen, DOI: 10.5772/intechopen.75300. Available from:

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