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Local Discontinuous Galerkin Method for Nonlinear Ginzburg- Landau Equation

Written By

Tarek Aboelenen

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

DOI: 10.5772/intechopen.75300

From the Edited Volume

Differential Equations - Theory and Current Research

Edited by Terry E. Moschandreou

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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,γ are 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.

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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,s and 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 Ω with boundary ∂Ω and assume that a nonoverlapping element Dk such 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 L2Dk norm

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 Ω into K nonoverlapping elements, Dk=xk12xk+12, Δxk=xk+12xk12 and k=1,,K. We assume uh,rh,shVkN be the approximation of u,r,s respectively, where the approximation space is defined as

VkN=v:vkPNDkDkΩ,E10

where PNDk denotes 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 n̂ is simply a scalar and takes the value of +1 and −1 at the right and the left interface, respectively.

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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=uhνuhνsh in (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 P and P+ into Vhk. For all the elements, Dk, k=1,2,,K are 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 uh be 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. ⃞.

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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 uh is 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 uhn to uhn+1, separated by the time step, Δt. In our examples, the condition ΔtCΔxminα0<C<1 is 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=axeidlnaxiωt where

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+1 order 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 uhxt for γ=2,1,0,1,2 is shown in Figures 2 and 3. The parameter γ will affect the wave shape. From these figures, it is obvious that the solution decays rapidly with time evolution especially for γ<0 and the parameter γ dramatically 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.

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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.

References

  1. 1. Aranson IS, Kramer L. The world of the complex Ginzburg-Landau equation. Reviews of Modern Physics. 2002;74:99
  2. 2. Stewartson K, Stuart J. A non-linear instability theory for a wave system in plane Poiseuille flow. Journal of Fluid Mechanics. 1971;48:529-545
  3. 3. Wang T, Guo B. Analysis of some finite difference schemes for two-dimensional Ginzburg-landau equation. Numerical Methods for Partial Differential Equations. 2011;27:1340-1363
  4. 4. Shokri A, Dehghan M. A meshless method using radial basis functions for the numerical solution of two-dimensional complex Ginzburg-Landau equation. Computer Modeling in Engineering and Sciences. 2012;84:333
  5. 5. Wang H. An efficient Chebyshev–Tau spectral method for Ginzburg–Landau–Schrödinger equations. Computer Physics Communications. 2010;181:325-340
  6. 6. Chen Z. Mixed finite element methods for a dynamical Ginzburg-Landau model in superconductivity. Numerische Mathematik. 1997;76:323-353
  7. 7. Gao H, Li B, Sun W. Optimal error estimates of linearized Crank-Nicolson Galerkin FEMs for the time-dependent Ginzburg–Landau equations in superconductivity. SIAM Journal on Numerical Analysis. 2014;52:1183-1202
  8. 8. Wang S, Zhang L. An efficient split-step compact finite difference method for cubic–quintic complex Ginzburg–Landau equations. Computer Physics Communications. 2013;184:1511-1521
  9. 9. Wang T, Guo B. A robust semi-explicit difference scheme for the Kuramoto–Tsuzuki equation. Journal of Computational and Applied Mathematics. 2009;233:878-888
  10. 10. Hao Z-P, Sun Z-Z, Cao W-R. A three-level linearized compact difference scheme for the Ginzburg–Landau equation. Numerical Methods for Partial Differential Equations. 2015;31:876-899
  11. 11. Shokri A, Afshari F. High-order compact ADI method using predictor–corrector scheme for 2D complex Ginzburg–landau equation. Computer Physics Communications. 2015;197:43-50
  12. 12. Arnold DN, Brezzi F, Cockburn B, Marini LD. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM Journal on Numerical Analysis. 2002;39:1749-1779
  13. 13. Hesthaven JS, Warburton T. Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. 1st ed. New York: Springer Publishing Company, Incorporated; 2007
  14. 14. Bernardo Cockburn C-WS, Karniadakis GE. Discontinuous Galerkin Methods: Theory, Computation and Applications. 1st ed. New York: Springer; 2000
  15. 15. El-Tantawy S, Aboelenen T. Simulation study of planar and nonplanar super rogue waves in an electronegative plasma: Local discontinuous Galerkin method. Physics of Plasmas. 2017;24:052118
  16. 16. Yan J, Shu C-W. Local discontinuous Galerkin methods for partial differential equations with higher order derivatives. Journal of Scientific Computing. 2002;17:27-47
  17. 17. Aboelenen T. Local discontinuous Galerkin method for distributed-order time and space-fractional convection–diffusion and Schrödinger-type equations. Nonlinear Dynamics. 2017:1-19
  18. 18. Aboelenen T. A high-order nodal discontinuous Galerkin method for nonlinear fractional Schrödinger type equations. Communications in Nonlinear Science and Numerical Simulation. 2018;54:428-452
  19. 19. Aboelenen T, El-Hawary H. A high-order nodal discontinuous Galerkin method for a linearized fractional Cahn–Hilliard equation. Computers & Mathematics with Applications. 2017;73:1197-1217
  20. 20. Aboelenen T. Discontinuous Galerkin methods for fractional elliptic problems; 2018. arXiv preprint arXiv:1802.02327
  21. 21. Ciarlet PG. Finite Element Method for Elliptic Problems. Philadelphia, PA, USA: Society for Industrial and Applied Mathematics; 2002
  22. 22. Cockburn B. High-Order Methods for Computational Physics, Berlin, Heidelberg: Springer; 1999. pp. 69-224. DOI: 10.1007/978-3-662-03882-6_2
  23. 23. Gottlieb S, Shu C-W. Total variation diminishing Runge-Kutta schemes. Mathematics of Computation. 1998;67:73-85

Written By

Tarek Aboelenen

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