Local Discontinuous Galerkin Method for Nonlinear Ginzburg- Landau Equation

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.

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

$$\frac{\partial u}{\partial t}-\left(\nu +\mathit{i\eta }\right)\mathrm{\Delta }u+\left(\kappa +\mathit{i\zeta }\right){\left|u\right|}^{2}u-\mathit{\gamma u}=0,$$E1

and periodic boundary conditions and $\eta ,\zeta ,\gamma$ are real constants, $\nu ,\kappa >0$. 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 $r,s$ and set

$$r=\frac{\partial }{\partial x}s,s=\frac{\partial }{\partial x}u,$$E2

then, the Ginzburg-Landau problem can be rewritten as

$$\begin{array}{l}\frac{\partial u}{\partial t}-\left(\nu +\mathit{i\eta }\right)r+\left(\kappa +\mathit{i\zeta }\right){\left|u\right|}^{2}u-\mathit{\gamma u}=0,\\ r=\frac{\partial }{\partial x}s,s=\frac{\partial }{\partial x}u.\end{array}$$E3

We consider problem posed on the physical domain $\mathrm{\Omega }$ with boundary $\mathrm{\partial \Omega }$ and assume that a nonoverlapping element $D^k$ such that

$$\mathrm{\Omega }=\underset{k=1}{\overset{K}{\cup }}{D}^k.$$E4

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

$${\left.H^r\left(\mathrm{\Omega }\right)=\{v\in L^2\left(\mathrm{\Omega }\right):\forall k=1,2,.\dots K,v\right|}_{D^k}\in H^r\left(D^k\right)\}.$$E5

We define the local inner product and $L^2\left(D^k\right)$ norm

$${\left(u,v\right)}_{D^k}={\int }_{D^k}\mathit{\text{uvdx}},\parallel u{\parallel }_{D^k}^2={\left(u,u\right)}_{D^k},$$E6

as well as the global broken inner product and norm

$${\left(u,v\right)}_{\mathrm{\Omega }}=\sum _{k=1}^K{\left(u,v\right)}_{D^k},\parallel u{\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2=\sum _{k=1}^K{\left(u,u\right)}_{D^k}.$$E7

We define the jumps along a normal, $\widehat{n}$, as

$$\left[u\right]={\widehat{n}}^{-}{u}^{-}+{\widehat{n}}^{+}{u}^{+}.$$E8

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

$$u^{\ast }=\left\{u\right\}=\frac{u^{+}+u^{-}}{2},s^{\ast }=\left\{s\right\}=\frac{s^{+}+s^{-}}{2}.$$E9

Let us discretize the computational domain $\mathrm{\Omega }$ into $K$ nonoverlapping elements, $D^k=\left[x_{k-\frac{1}{2}},x_{k+\frac{1}{2}}\right]$, $\mathrm{\Delta }{x}_k=x_{k+\frac{1}{2}}-x_{k-\frac{1}{2}}$ and $k=1,\dots ,K$. We assume $u_h,r_h,s_h\in V_k^N$ be the approximation of $u,r,s$ respectively, where the approximation space is defined as

$$V_k^N=\left\{v:v_k\in {\mathbb{P}}^N\left(D^k\right),\forall D^k\in \mathrm{\Omega }\right\},$$E10

where ${\mathbb{P}}^N\left(D^k\right)$ denotes the set of polynomials of degree up to N defined on the element $D^k$. We define local discontinuous Galerkin scheme as follows: find $u_h,r_h,s_h\in V_k^N$, such that for all test functions $\vartheta ,\varphi ,\phi \in V_k^N$,

$$\begin{array}{l}{\left(\frac{\partial u_h}{\partial t},\vartheta \right)}_{D^k}-\left(\nu +\mathit{i\eta }\right){\left(r_h,\vartheta \right)}_{D^k}+\left(\kappa +\mathit{i\zeta }\right){\left({\left|u_h\right|}^2{u}_h,\vartheta \right)}_{D^k}-\gamma {\left(u_h,\vartheta \right)}_{D^k}=0,\\ {\left(r_h,\varphi \right)}_{D^k}={\left(\frac{\partial }{\partial x}{s}_h,\varphi \right)}_{D^k},\\ {\left(s_h,\phi \right)}_{D^k}={\left(\frac{\partial }{\partial x}{u}_h,\phi \right)}_{D^k}.\end{array}$$E11

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

$$\begin{array}{l}{\left(\frac{\partial u_h}{\partial t},\vartheta \right)}_{D^k}-\left(\nu +\mathit{i\eta }\right){\left(r_h,\vartheta \right)}_{D^k}+\left(\kappa +\mathit{i\zeta }\right){\left({\left|u_h\right|}^2{u}_h,\vartheta \right)}_{D^k}-\gamma {\left(u_h,\vartheta \right)}_{D^k}=0,\\ {\left(r_h,\varphi \right)}_{D^k}=-{\left(s_h,{\varphi }_x\right)}_{D^k}+{\left(s_h^{\ast }{\varphi }^{-}\right)}_{k+\frac{1}{2}}-{\left(s_h^{\ast }{\varphi }^{+}\right)}_{k-\frac{1}{2}},\\ {\left(s_h,\phi \right)}_{D^k}=-{\left(u_h,{\phi }_x\right)}_{D^k}+{\left(u_h^{\ast }{\phi }^{-}\right)}_{k+\frac{1}{2}}-{\left(u_h^{\ast }{\phi }^{+}\right)}_{k-\frac{1}{2}},\end{array}$$E12

we can rewrite (12) as

$$\begin{array}{l}{\left(\frac{\partial u_h}{\partial t},\vartheta \right)}_{D^k}-\left(\nu +\mathit{i\eta }\right){\left(r_h,\vartheta \right)}_{D^k}+\left(\kappa +\mathit{i\zeta }\right){\left({\left|u_h\right|}^2{u}_h,\vartheta \right)}_{D^k}-\gamma {\left(u_h,\vartheta \right)}_{D^k}=0,\\ {\left(r_h,\varphi \right)}_{D^k}=-{\left(s_h,{\varphi }_x\right)}_{D^k}+{\left(\widehat{n}.s_h^{\ast },\varphi \right)}_{\partial D^k},\\ {\left(s_h,\phi \right)}_{D^k}=-{\left(u_h,{\phi }_x\right)}_{D^k}+{\left(\widehat{n}.u_h^{\ast },\phi \right)}_{\partial D^k}.\end{array}$$E13

where $\widehat{n}$ is 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

$$\parallel u_h\left(x,T\right){\parallel }_{\mathrm{\Omega }}\le e^{-2\mathit{\gamma T}}\parallel u_0\left(x\right){\parallel }_{\mathrm{\Omega }}$$

for any $T>0$.

Proof. Set $\left(\vartheta ,\varphi ,\phi \right)=\left(u_h,\nu u_h,\nu s_h\right)$ in (13) and consider the integration by parts formula ${\left(u\text{,}\frac{\partial r}{\partial x}\right)}_{D^k}+{\left(r\text{,}\frac{\partial u}{\partial x}\right)}_{D^k}={\left[ur\right]}_{{}_{x_{k-\frac{1}{2}}}}^{{}_{x_{k+\frac{1}{2}}}}$, we get

$$\begin{array}{l}{\left({\left(u_h\right)}_t,u_h\right)}_{D^k}+{\left(s_h,s_h\right)}_{D^k}\\ =-\nu {\left(r_h,u_h\right)}_{D^k}+\left(\nu +\mathit{i\eta }\right){\left(r_h,u_h\right)}_{D^k}-\left(\kappa +\mathit{i\zeta }\right){\left({\left|u_h\right|}^2{u}_h,u_h\right)}_{D^k}\\ +\gamma {\left(u_h,u_h\right)}_{D^k}+\nu {\left(\widehat{n}.s_h^{\ast },u_h\right)}_{\partial D^k}+\nu {\left(\widehat{n}.u_h^{\ast },s_h\right)}_{\partial D^k}-\nu {\left(\widehat{n}.s_h,u_h\right)}_{\partial D^k}.\end{array}$$E14

Taking the real part of the resulting equation, we obtain

$$\begin{array}{ll}{\left({\left(u_h\right)}_t,u_h\right)}_{D^k}+{\left(s_h,s_h\right)}_{D^k}=& -\kappa {\left({\left|u_h\right|}^2{u}_h,u_h\right)}_{D^k}+\gamma {\left(u_h,u_h\right)}_{D^k}\hfill \\ & +\nu {\left(\widehat{n}.s_h^{\ast },u_h\right)}_{\partial D^k}+\nu {\left(\widehat{n}.u_h^{\ast },s_h\right)}_{\partial D^k}-\nu {\left(\widehat{n}.s_h,u_h\right)}_{\partial D^k}.\hfill \end{array}$$E15

Removing the positive term $\kappa {\left({\left|u_h\right|}^2{u}_h,u_h\right)}_{D^k}$, we obtain

$${\left({\left(u_h\right)}_t,u_h\right)}_{D^k}+{\left(s_h,s_h\right)}_{D^k}\le \gamma \parallel u_h{\parallel }_{L^2\left(D^k\right)}^2+\nu {\left(\widehat{n}.s_h^{\ast },u_h\right)}_{\partial D^k}+\nu {\left(\widehat{n}.u_h^{\ast },s_h\right)}_{\partial D^k}-\nu {\left(\widehat{n}.s_h,u_h\right)}_{\partial D^k}.$$E16

Summing over all elements (16), we easily obtain

$${\left({\left(u_h\right)}_t,u_h\right)}_{L^2\left(\mathrm{\Omega }\right)}+{\left(s_h,s_h\right)}_{L^2\left(\mathrm{\Omega }\right)}\le \gamma \parallel u_h{\parallel }_{\mathrm{\Omega }}^2.$$E17

Employing Gronwall’s inequality, we obtain

$$\parallel u_h\left(x,T\right){\parallel }_{\mathrm{\Omega }}\le e^{-2\mathit{\gamma T}}\parallel u_0\left(x\right){\parallel }_{\mathrm{\Omega }}.\square$$

3.2. Error estimates

We consider the linear Ginzburg-Landau equation

$$\frac{\partial u}{\partial t}-\left(\nu +\mathit{i\eta }\right)\mathrm{\Delta }u+\left(\kappa +\mathit{i\zeta }\right)u-\mathit{\gamma u}=0.$$E18

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

$$\begin{array}{l}{\left(u_t,\vartheta \right)}_{D^k}-\left(\nu +\mathit{i\eta }\right){\left(r,\vartheta \right)}_{D^k}+\left(\kappa +\mathit{i\zeta }\right){\left(u,\vartheta \right)}_{D^k}-\gamma {\left(u,\vartheta \right)}_{D^k}=0,\\ {\left(r,\varphi \right)}_{D^k}=-{\left(s,{\varphi }_x\right)}_{D^k}+{\left(\widehat{n}.s^{\ast },\varphi \right)}_{\partial D^k},\\ {\left(s,\phi \right)}_{D^k}=-{\left(u,{\phi }_x\right)}_{D^k}+{\left(\widehat{n}.u^{\ast },\phi \right)}_{\partial D^k}.\end{array}$$E19

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

$$\begin{array}{ll}\hfill & {\left({\left(u-u_h\right)}_t,\vartheta \right)}_{D^k}+{\left(s-s_h,{\varphi }_x\right)}_{D^k}+{\left(u-u_h,{\phi }_x\right)}_{D^k}+\left(\kappa +\mathit{i\zeta }\right){\left(u-u_h,\vartheta \right)}_{D^k}\\ & -\gamma {\left(u-u_h,\vartheta \right)}_{D^k}+{\left(r-r_h,\varphi \right)}_{D^k}+{\left(s-s_h,\phi \right)}_{D^k}-{\left(\widehat{n}.{\left(s-s_h\right)}^{\ast },\varphi \right)}_{\partial D^k}\hfill \\ & -\left(\nu +\mathit{i\eta }\right){\left(r-r_h,\vartheta \right)}_{D^k}-{\left(\widehat{n}.{\left(u-u_h\right)}^{\ast },\phi \right)}_{\partial D^k}=0.\hfill \end{array}$$E20

For the error estimate, we define special projections ${\mathcal{P}}^{-}$ and ${\mathcal{P}}^{+}$ into $V_h^k$. For all the elements, $D^k$, $k=1,2,\dots ,K$ are defined to satisfy

$$\begin{array}{c}{\left({\mathcal{P}}^{+}u-u,v\right)}_{D^k}=0,\forall v\in {\mathbb{P}}_N^k\left(D^k\right),{\mathcal{P}}^{+}u\left(x_{k-\frac{1}{2}}\right)=u\left(x_{k-\frac{1}{2}}\right),\\ {\left({\mathcal{P}}^{-}u-u,v\right)}_{D^k}=0,\forall v\in {\mathbb{P}}_N^{k-1}\left(D^k\right),{\mathcal{P}}^{-}u\left(x_{k+\frac{1}{2}}\right)=u\left(x_{k+\frac{1}{2}}\right).\end{array}$$E21

Denoting

$$\begin{array}{l}\pi ={\mathcal{P}}^{-}u-u_h,{\pi }^e={\mathcal{P}}^{-}u-u,\epsilon ={\mathcal{P}}^{+}r-r_h,{\epsilon }^e={\mathcal{P}}^{+}r-r,\\ \tau ={\mathcal{P}}^{+}s-s_h,{\tau }^e={\mathcal{P}}^{+}s-s.\end{array}$$E22

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

$$\begin{array}{c}\parallel {\mathcal{P}}^{+}u\left(.\right)-u\left(.\right){\parallel }_{L^2\left({\mathrm{\Omega }}_h\right)}\le {\mathit{Ch}}^{N+1},\\ \parallel {\mathcal{P}}^{-}u\left(.\right)-u\left(.\right){\parallel }_{L^2\left({\mathrm{\Omega }}_h\right)}\le {\mathit{Ch}}^{N+1},\end{array}$$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 $u_h$ be the numerical solution of the semi-discrete LDG scheme (13). Then for small enough h, we have the following error estimates:

$$\parallel u\left(.,t\right)-u_h\left(.,t\right){\parallel }_{L^2\left({\mathrm{\Omega }}_h\right)}\le {\mathit{Ch}}^{N+1},$$E24

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

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

$$\begin{array}{ll}\hfill & {\left({\left(\pi -{\pi }^e\right)}_t,\vartheta \right)}_{D^k}+{\left(\tau -{\tau }^e,{\varphi }_x\right)}_{D^k}+{\left(\pi -{\pi }^e,{\phi }_x\right)}_{D^k}+\left(\kappa +\mathit{i\zeta }\right){\left(\pi -{\pi }^e,\vartheta \right)}_{D^k}-\gamma {\left(\pi -{\pi }^e,\vartheta \right)}_{D^k}\\ & +{\left(\epsilon -{\epsilon }^e,\varphi \right)}_{D^k}+{\left(\tau -{\tau }^e,\phi \right)}_{D^k}+{\left(\varphi -{\varphi }^e,\beta \right)}_{D^k}-{\left(\widehat{n}.{\left(\tau -{\tau }^e\right)}^{\ast },\varphi \right)}_{\partial D^k}-\left(\nu +\mathit{i\eta }\right)\hfill \\ & {\left(\epsilon -{\epsilon }^e,\vartheta \right)}_{D^k}-{\left(\widehat{n}.{\left(\pi -{\pi }^e\right)}^{\ast },\phi \right)}_{\partial D^k}=0.\hfill \end{array}$$E25

Taking the real part of the resulting equation, we obtain

$$\begin{array}{ll}\hfill & {\left({\left(\pi -{\pi }^e\right)}_t,\vartheta \right)}_{D^k}+{\left(\tau -{\tau }^e,{\varphi }_x\right)}_{D^k}+{\left(\pi -{\pi }^e,{\phi }_x\right)}_{D^k}+\kappa {\left(\pi -{\pi }^e,\vartheta \right)}_{D^k}\\ & -\gamma {\left(\pi -{\pi }^e,\vartheta \right)}_{D^k}+{\left(\epsilon -{\epsilon }^e,\varphi \right)}_{D^k}+{\left(\tau -{\tau }^e,\phi \right)}_{D^k}-{\left(\widehat{n}.{\left(\tau -{\tau }^e\right)}^{\ast },\varphi \right)}_{\partial D^k}\hfill \\ & -\nu {\left(\epsilon -{\epsilon }^e,\vartheta \right)}_{D^k}-{\left(\widehat{n}.{\left(\pi -{\pi }^e\right)}^{\ast },\phi \right)}_{\partial D^k}=0.\hfill \end{array}$$E26

We take the test functions

$$\vartheta =\pi ,\varphi =\mathit{\nu \pi },\phi =\mathit{\nu \tau },$$E27

we obtain

$$\begin{array}{ll}\hfill & {\left({\left(\pi -{\pi }^e\right)}_t,\pi \right)}_{D^k}+\nu {\left(\tau -{\tau }^e,{\pi }_x\right)}_{D^k}+\nu {\left(\pi -{\pi }^e,{\tau }_x\right)}_{D^k}\\ & +\kappa {\left(\pi -{\pi }^e,\pi \right)}_{D^k}-\gamma {\left(\pi -{\pi }^e,\pi \right)}_{D^k}+\nu {\left(\epsilon -{\epsilon }^e,\pi \right)}_{D^k}\hfill \\ & +\nu {\left(\tau -{\tau }^e,\tau \right)}_{D^k}-\nu {\left(\widehat{n}.{\left(\tau -{\tau }^e\right)}^{\ast },\pi \right)}_{\partial D^k}-\nu {\left(\epsilon -{\epsilon }^e,\pi \right)}_{D^k}-\nu {\left(\widehat{n}.{\left(\pi -{\pi }^e\right)}^{\ast },\tau \right)}_{\partial D^k}=0.\hfill \end{array}$$E28

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

$$\begin{array}{l}{\left({\pi }_t,\pi \right)}_{\mathrm{\Omega }}+\nu {\left(\tau ,\tau \right)}_{\mathrm{\Omega }}=\nu {\left({\tau }^e,{\pi }_x\right)}_{\mathrm{\Omega }}+\nu {\left({\pi }^e,{\tau }_x\right)}_{\mathrm{\Omega }}+{\left({\pi }_t^e,\pi \right)}_{\mathrm{\Omega }}-\gamma {\left({\pi }^e,\pi \right)}_{\mathrm{\Omega }}+\kappa {\left({\pi }^e,\pi \right)}_{\mathrm{\Omega }}\\ +\nu {\left({\tau }^e,\tau \right)}_{\mathrm{\Omega }}+\gamma {\left(\pi ,\pi \right)}_{\mathrm{\Omega }}-\kappa {\left(\pi ,\pi \right)}_{\mathrm{\Omega }}-\nu \sum _{k=1}^K{\left(\widehat{n}.{\left({\pi }^e\right)}^{\ast },\tau \right)}_{\partial D^k}-\nu \sum _{k=1}^K{\left(\widehat{n}.{\left({\tau }^e\right)}^{\ast },\pi \right)}_{\partial D^k},\end{array}$$E29

we can rewrite (29) as

$${\left({\pi }_t,\pi \right)}_{\mathrm{\Omega }}+\nu {\left(\tau ,\tau \right)}_{\mathrm{\Omega }}=I+\mathit{II}+\mathit{III},$$E30

where

$$I=\nu {\left({\tau }^e,{\pi }_x\right)}_{\mathrm{\Omega }}+\nu {\left({\pi }^e,{\tau }_x\right)}_{\mathrm{\Omega }},$$E31

$$\mathit{II}={\left({\pi }_t^e,\pi \right)}_{\mathrm{\Omega }}-\gamma {\left({\pi }^e,\pi \right)}_{\mathrm{\Omega }}+\kappa {\left({\pi }^e,\pi \right)}_{\mathrm{\Omega }}+\nu {\left({\tau }^e,\tau \right)}_{\mathrm{\Omega }}-\nu \sum _{k=1}^K{\left(\widehat{n}.{\left({\pi }^e\right)}^{\ast },\tau \right)}_{\partial D^k}-\nu \sum _{k=1}^K{\left(\widehat{n}.{\left({\tau }^e\right)}^{\ast },\pi \right)}_{\partial D^k},$$E32

$$\mathit{III}=\gamma {\left(\pi ,\pi \right)}_{\mathrm{\Omega }}-\kappa {\left(\pi ,\pi \right)}_{\mathrm{\Omega }}.$$E33

Using the definitions of the projections $\mathcal{P},\mathcal{S}$ (21) in (31), we get

$$\begin{array}{l}I=0.\end{array}$$E34

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

$$\mathit{II}\le c_1\parallel \pi {\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2+c_2\parallel \tau {\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2+{\mathit{Ch}}^{2N+2}.$$E35

and

$$\mathit{III}\le c_1\parallel \pi {\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2.$$E36

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

$${\left({\pi }_t,\pi \right)}_{\mathrm{\Omega }}+\nu {\left(\tau ,\tau \right)}_{\mathrm{\Omega }}\le c_1\parallel \pi {\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2+c_2\parallel \tau {\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2+{\mathit{Ch}}^{2N+2},$$E37

provided c₂ is sufficiently small such that $c_2\le \nu$, we obtain that

$${\left({\pi }_t,\pi \right)}_{\mathrm{\Omega }}\le c_1\parallel \pi {\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2+{\mathit{Ch}}^{2N+2}.$$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

$$\begin{array}{l}\frac{\partial {\mathbf{u}}_h}{\partial t}=\mathcal{F}\left({\mathbf{u}}_h,t\right),\end{array}$$E39

where ${\mathbf{u}}_h$ is the vector of unknowns, we can use the standard fourth-order four-stage explicit RK method (ERK)

$$\begin{array}{l}{\mathbf{k}}^1=\mathcal{F}\left({\mathbf{u}}_h^n,t^n\right),\\ {\mathbf{k}}^2=\mathcal{F}\left({\mathbf{u}}_h^n+\frac{1}{2}\mathrm{\Delta }t{\mathbf{k}}^1,t^n+\frac{1}{2}\mathrm{\Delta }t\right),\\ {\mathbf{k}}^3=\mathcal{F}\left({\mathbf{u}}_h^n+\frac{1}{2}\mathrm{\Delta }t{\mathbf{k}}^2,t^n+\frac{1}{2}\mathrm{\Delta }t\right),\\ {\mathbf{k}}^4=\mathcal{F}\left({\mathbf{u}}_h^n+\mathrm{\Delta }t{\mathbf{k}}^3,t^n+\mathrm{\Delta }t\right),\\ {\mathbf{u}}_h^{n+1}={\mathbf{u}}_h^n+\frac{1}{6}\left({\mathbf{k}}^1+2{\mathbf{k}}^2+2{\mathbf{k}}^3+{\mathbf{k}}^4\right),\end{array}$$E40

to advance from ${\mathbf{u}}_h^n$ to ${\mathbf{u}}_h^{n+1}$, separated by the time step, $\mathrm{\Delta }t$. In our examples, the condition $\mathrm{\Delta }t\le C\mathrm{\Delta }{x}_{\mathit{min}}^{\alpha }\left(0<C<1\right)$ is used to ensure stability.

Example 4.1 We consider the following linear Ginzburg-Landau equation

$$\frac{\partial u}{\partial t}-\left(\nu +\mathit{i\eta }\right)\mathrm{\Delta }u+\left(\kappa +\mathit{i\zeta }\right)u=0,x\in \left[-20,20\right],t\in \left(\mathrm{0,0.5}\right],u\left(x,0\right)=u_0\left(x\right),$$E41

with

$$\begin{array}{l}\eta =\frac{1}{2},\kappa =-\frac{\nu \left(3\sqrt{1+4{\nu }^2}-1\right)}{2\left(2+9{\nu }^2\right)},\zeta =-1,\gamma =0.\end{array}$$E42

The exact solution $u\left(x,t\right)=a\left(x\right)e^{\mathit{id}ln\left(a\left(x\right)\right)-\mathit{i\omega t}}$ where

$$\begin{array}{l}a\left(x\right)=\mathit{\text{Fsech}}\left(x\right),F=\sqrt{\frac{d\sqrt{1+4{\nu }^2}}{-2\kappa }},d=\frac{\sqrt{1+4{\nu }^2}-1}{2\nu },\omega =-\frac{d\left(1+4{\nu }^2\right)}{2\nu }.\end{array}$$E43

The convergence rates and the numerical L² error are listed in Figure 1 for several different values of $\nu$, confirming optimal $O\left(h^{N+1}\right)$ order of convergence across.

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

$$\begin{array}{l}u\left(x,0\right)=e^{-x^2},\end{array}$$E44

with parameters $\nu =1,\kappa =1,\eta =1,\zeta =2$, $x\in \left[-10,10\right]$. We consider cases with N = 2 and K = 40 and solve the equation for several different values of $\gamma$. The numerical solution $u_h\left(x,t\right)$ for $\gamma =2,1,0,-1,-2$ is shown in Figures 2 and 3. The parameter $\gamma$ will affect the wave shape. From these figures, it is obvious that the solution decays rapidly with time evolution especially for $\gamma <0$ and the parameter $\gamma$ 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.

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.