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

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

$$\frac{\partial \mu}{\partial t} - (\nu + i\eta)\Delta u + (\kappa + i\zeta)|u|^2 u - \gamma u = 0,\tag{1}$$

Hr

We define the local inner product and L<sup>2</sup> Dk � � norm

as well as the global broken inner product and norm

<sup>2</sup> � xk�<sup>1</sup>

k ,

Dk <sup>¼</sup> <sup>∂</sup>

corresponding numerical fluxes, we obtain

ð Þ sh; φ Dk ¼ � uh;φ<sup>x</sup>

Dk ¼ � sh; ϕ<sup>x</sup> � �

� �

ð Þ sh; <sup>φ</sup> Dk <sup>¼</sup> <sup>∂</sup>

<sup>∂</sup><sup>x</sup> sh; <sup>ϕ</sup> � �

<sup>∂</sup><sup>x</sup> uh;<sup>φ</sup> � �

We define the jumps along a normal, n^, as

xk�<sup>1</sup> 2 ; xkþ<sup>1</sup> 2 h i, <sup>Δ</sup>xk <sup>¼</sup> xkþ<sup>1</sup>

functions ϑ, ϕ,φ ∈V<sup>N</sup>

∂uh <sup>∂</sup><sup>t</sup> ; <sup>ϑ</sup> � �

∂uh <sup>∂</sup><sup>t</sup> ; <sup>ϑ</sup> � �

we can rewrite (12) as

rh; ϕ � �

rh; ϕ � �

ð Þ¼f <sup>Ω</sup> <sup>v</sup> <sup>∈</sup>L<sup>2</sup>

ð Þ u; v Dk ¼

K

k¼1

<sup>u</sup><sup>∗</sup> <sup>¼</sup> f g<sup>u</sup> <sup>¼</sup> <sup>u</sup><sup>þ</sup> <sup>þ</sup> <sup>u</sup>�

of u, r, s respectively, where the approximation space is defined as V<sup>N</sup>

define local discontinuous Galerkin scheme as follows: find uh, rh, sh ∈V<sup>N</sup>

<sup>D</sup><sup>k</sup> � ð Þ ν þ iη ð Þ rh; ϑ Dk þ ð Þ κ þ iζ j j uh

<sup>D</sup><sup>k</sup> � ð Þ ν þ iη ð Þ rh; ϑ Dk þ ð Þ κ þ iζ j j uh

Dk <sup>þ</sup> <sup>s</sup><sup>∗</sup> hϕ� � � <sup>k</sup>þ<sup>1</sup> 2 � <sup>s</sup><sup>∗</sup> hϕ<sup>þ</sup> � � <sup>k</sup>�<sup>1</sup> 2 ,

<sup>D</sup><sup>k</sup> <sup>þ</sup> <sup>u</sup><sup>∗</sup> hφ� � �

Dk ,

Dk :

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

ð Þ <sup>u</sup>; <sup>v</sup> <sup>Ω</sup> <sup>¼</sup> <sup>X</sup>

ð Dk

ð Þ Ω : ∀k ¼ 1; 2; :…K; vj

uvdx, ∥u∥<sup>2</sup>

ð Þ <sup>u</sup>; <sup>v</sup> <sup>D</sup><sup>k</sup> , <sup>∥</sup>u∥<sup>2</sup>

<sup>L</sup>2ð Þ <sup>Ω</sup> <sup>¼</sup> <sup>X</sup> K

<sup>2</sup> , s<sup>∗</sup> <sup>¼</sup> f g<sup>s</sup> <sup>¼</sup> <sup>s</sup><sup>þ</sup> <sup>þ</sup> <sup>s</sup>�

2 uh; ϑ � �

2 uh; ϑ � �

> <sup>k</sup>�<sup>1</sup> 2 ,

Let us discretize the computational domain <sup>Ω</sup> into <sup>K</sup> nonoverlapping elements, <sup>D</sup><sup>k</sup> <sup>¼</sup>

where P<sup>N</sup> D<sup>k</sup> � � denotes the set of polynomials of degree up to N defined on the element D<sup>k</sup>

Applying integration by parts to (11), and replacing the fluxes at the interfaces by the

<sup>k</sup>þ<sup>1</sup> 2 � <sup>u</sup><sup>∗</sup> hφ<sup>þ</sup> � �

<sup>2</sup> and <sup>k</sup> <sup>¼</sup> <sup>1</sup>, …, K. We assume uh, rh, sh <sup>∈</sup> <sup>V</sup><sup>N</sup>

k¼1

Local Discontinuous Galerkin Method for Nonlinear Ginzburg-Landau Equation

½ �¼ u n^�u� þ n^þuþ: (8)

<sup>k</sup> <sup>¼</sup> <sup>v</sup> : vk <sup>∈</sup>P<sup>N</sup> <sup>D</sup><sup>k</sup> � �; <sup>∀</sup>D<sup>k</sup> <sup>∈</sup> <sup>Ω</sup> � �, (10)

<sup>D</sup><sup>k</sup> � <sup>γ</sup>ð Þ uh; <sup>ϑ</sup> <sup>D</sup><sup>k</sup> <sup>¼</sup> <sup>0</sup>,

<sup>D</sup><sup>k</sup> � <sup>γ</sup>ð Þ uh; <sup>ϑ</sup> <sup>D</sup><sup>k</sup> <sup>¼</sup> <sup>0</sup>,

Dk <sup>∈</sup> <sup>H</sup><sup>r</sup> <sup>D</sup><sup>k</sup> � �g: (5)

http://dx.doi.org/10.5772/intechopen.75300

119

ð Þ u; u Dk : (7)

<sup>2</sup> : (9)

<sup>k</sup> be the approximation

<sup>k</sup> , such that for all test

. We

(11)

(12)

Dk ¼ ð Þ u; u Dk , (6)

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 [3–11]. The local discontinuous Galerkin (LDG) method is famous for high accuracy properties and extreme flexibility [12–20]. 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<sup>2</sup> 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.
