## 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

and periodic boundary conditions and

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 *L*^{2} 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

then, the Ginzburg-Landau problem can be rewritten as

We consider problem posed on the physical domain

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

We define the local inner product and

as well as the global broken inner product and norm

We define the jumps along a normal,

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

Let us discretize the computational domain

where *N* defined on the element

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

we can rewrite (12) as

where

## 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.** (*L*^{2} *stability). The solution given by the LDG method defined by* (13) *satisfies*

*for any*

**Proof.** Set

Taking the real part of the resulting equation, we obtain

Removing the positive term

Summing over all elements (16), we easily obtain

Employing Gronwall’s inequality, we obtain

### 3.2. Error estimates

We consider the linear Ginzburg-Landau equation

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

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

For the error estimate, we define special projections

Denoting

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

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* *be the numerical solution of the semi-discrete LDG scheme* (13)*. Then for small enough h, we have the following error estimates:*

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

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

Taking the real part of the resulting equation, we obtain

We take the test functions

we obtain

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

we can rewrite (29) as

where

Using the definitions of the projections

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

and

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

provided *c*_{2} is sufficiently small such that

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

where

to advance from

**Example 4.1** *We consider the following linear Ginzburg-Landau equation*

with

The exact solution

The convergence rates and the numerical *L*^{2} error are listed in Figure 1 for several different values of

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

with parameters *N* = 2 and *K* = 40 and solve the equation for several different values of

## 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.