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

$$r = \frac{\partial}{\partial x} s, \quad s = \frac{\partial}{\partial x} u,\tag{2}$$

then, the Ginzburg-Landau problem can be rewritten as

$$\begin{aligned} \frac{\partial \mu}{\partial t} - (\nu + i\eta)r + (\kappa + i\zeta)|u|^2 u - \gamma u &= 0, \\ r = \frac{\partial}{\partial x} s, \quad s = \frac{\partial}{\partial x} u. \end{aligned} \tag{3}$$

We consider problem posed on the physical domain Ω with boundary ∂Ω and assume that a nonoverlapping element D<sup>k</sup> such that

$$
\Omega = \bigcup\_{k=1}^{K} D^k. \tag{4}
$$

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

Local Discontinuous Galerkin Method for Nonlinear Ginzburg-Landau Equation http://dx.doi.org/10.5772/intechopen.75300 119

$$H^r(\Omega) = \{ \upsilon \in L^2(\Omega) : \forall k = 1, 2, \ldots K, \upsilon|\_{D^k} \in H^r(\mathcal{D}^k) \}. \tag{5}$$

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

∂u ∂t

118 Differential Equations - Theory and Current Research

concluding remarks are given in Section 5.

2. LDG scheme for Ginzburg-Landau equation

fluxes are used. We introduce variables r, s and set

nonoverlapping element D<sup>k</sup> such that

then, the Ginzburg-Landau problem can be rewritten as ∂u ∂t

> <sup>r</sup> <sup>¼</sup> <sup>∂</sup> ∂x

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

� ð Þ ν þ iη Δu þ ð Þ κ þ iζ j j u

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

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

> s, s <sup>¼</sup> <sup>∂</sup> ∂x

We consider problem posed on the physical domain Ω with boundary ∂Ω and assume that a

Ω ¼ ∪ K k¼1 Dk 2

u � γu ¼ 0,

<sup>r</sup> <sup>¼</sup> <sup>∂</sup> ∂x

� ð Þ ν þ iη r þ ð Þ κ þ iζ j j u

s, s <sup>¼</sup> <sup>∂</sup> ∂x u: 2

u � γu ¼ 0, (1)

u, (2)

: (4)

(3)

$$(u, v)\_{D^k} = \int\_{D^k} u v dx, \quad \|u\|\_{D^k}^2 = (u, u)\_{D^k} \tag{6}$$

as well as the global broken inner product and norm

$$(\boldsymbol{u}, \boldsymbol{v})\_{\Omega} = \sum\_{k=1}^{K} (\boldsymbol{u}, \boldsymbol{v})\_{\mathcal{D}^k} \quad \|\boldsymbol{u}\|\_{\boldsymbol{L}^2(\Omega)}^2 = \sum\_{k=1}^{K} (\boldsymbol{u}, \boldsymbol{u})\_{\mathcal{D}^k} \tag{7}$$

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

$$\left[\mu\right] = \hat{\mathfrak{n}}^{-}\hat{\mathfrak{u}}^{-} + \hat{\mathfrak{n}}^{+}\hat{\mathfrak{u}}^{+}.\tag{8}$$

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

$$\{\mu^\* = \{u\} = \frac{\mu^+ + \mu^-}{2}, \quad s^\* = \{s\} = \frac{s^+ + s^-}{2}. \tag{9}$$

Let us discretize the computational domain <sup>Ω</sup> into <sup>K</sup> nonoverlapping elements, <sup>D</sup><sup>k</sup> <sup>¼</sup> xk�<sup>1</sup> 2 ; xkþ<sup>1</sup> 2 h i, <sup>Δ</sup>xk <sup>¼</sup> xkþ<sup>1</sup> <sup>2</sup> � xk�<sup>1</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> <sup>k</sup> be the approximation of u, r, s respectively, where the approximation space is defined as

$$V\_k^N = \{ v : v\_k \in \mathbb{P}^N(\mathcal{D}^k), \forall \mathcal{D}^k \in \Omega \}, \tag{10}$$

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> . We define local discontinuous Galerkin scheme as follows: find uh, rh, sh ∈V<sup>N</sup> <sup>k</sup> , such that for all test functions ϑ, ϕ,φ ∈V<sup>N</sup> k ,

$$\begin{split} & \left( \frac{\partial u\_h}{\partial t}, \mathfrak{d} \right)\_{D^k} - (\nu + i\eta)(r\_h, \mathfrak{d})\_{D^k} + (\kappa + i\zeta) \left( |u\_h|^2 u\_h, \mathfrak{d} \right)\_{D^k} - \chi(u\_h, \mathfrak{d})\_{D^k} = 0, \\ & \left( r\_h, \phi \right)\_{D^k} = \left( \underset{\mathrm{div}}{\mathrm{d}} \, \_0\phi \right)\_{D^k} \\ & \left( \mathfrak{d}, \, \phi \right)\_{D^k} = \left( \underset{\mathrm{div}}{\mathrm{d}} u\_h, \phi \right)\_{D^k}. \end{split} \tag{11}$$

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

$$\begin{aligned} \left(\frac{\partial u\_h}{\partial t}, \mathfrak{d}\right)\_{D^k} - \left(\nu + i\eta\right) (r\_h, \mathfrak{d})\_{D^k} + \left(\kappa + i\zeta\right) \left(|u\_h|^2 u\_h, \mathfrak{d}\right)\_{D^k} - \gamma (u\_h, \mathfrak{d})\_{D^k} &= 0, \\ \left(r\_h, \phi\right)\_{D^k} &= -\left(s\_h, \phi\_x\right)\_{D^k} + \left(s\_h^\* \phi^-\right)\_{k + \frac{1}{2}} - \left(s\_h^\* \phi^+\right)\_{k - \frac{1}{2}} \\ \left(s\_h, \phi\right)\_{D^k} &= -\left(u\_h, \phi\_x\right)\_{D^k} + \left(u\_h^\* \phi^-\right)\_{k + \frac{1}{2}} - \left(u\_h^\* \phi^+\right)\_{k - \frac{1}{2}} \end{aligned} \tag{12}$$

we can rewrite (12) as

$$\begin{split} & \left( \frac{\partial \boldsymbol{u}}{\partial t}, \boldsymbol{\mathfrak{s}} \right)\_{D^{k}} - (\boldsymbol{\nu} + i\boldsymbol{\eta}) (\boldsymbol{r}\_{h}, \boldsymbol{\mathfrak{s}})\_{D^{k}} + (\boldsymbol{\kappa} + i\boldsymbol{\zeta}) \Big( |\boldsymbol{u}\_{h}|^{2} \boldsymbol{u}\_{h}, \boldsymbol{\mathfrak{s}} \Big)\_{D^{k}} - \boldsymbol{\chi} (\boldsymbol{u}\_{h}, \boldsymbol{\mathfrak{s}})\_{D^{k}} = \boldsymbol{0}, \\ & \Big( \boldsymbol{r}\_{h}, \boldsymbol{\phi} \big)\_{D^{k}} = - (\boldsymbol{s}\_{h}, \boldsymbol{\phi}\_{\boldsymbol{x}})\_{D^{k}} + \big( \boldsymbol{\hat{n}}, \boldsymbol{\mathfrak{s}}\_{h}^{\*}, \boldsymbol{\phi} \big)\_{\partial D^{k}}, \\ & \Big( \boldsymbol{s}\_{h}, \boldsymbol{\varphi} \big)\_{D^{k}} = - \big( \boldsymbol{u}\_{h}, \boldsymbol{\varphi}\_{\boldsymbol{x}} \big)\_{D^{k}} + \big( \boldsymbol{\hat{n}}, \boldsymbol{u}\_{h}^{\*}, \boldsymbol{\varphi} \big)\_{\partial D^{k}}. \end{split} \tag{13}$$

ð Þ uh <sup>t</sup> ; uh 

∥uhð Þ x; T ∥<sup>Ω</sup> ≤ e

Employing Gronwall's inequality, we obtain

We consider the linear Ginzburg-Landau equation

r; ϕ

ð Þ u � uh <sup>t</sup> ; ϑ

, k ¼ 1, 2, …, K are defined to satisfy

∂u ∂t

Dk ¼ � s; ϕ<sup>x</sup> 

ð Þ s;φ Dk ¼ � u; φ<sup>x</sup>

�γð Þ <sup>u</sup> � uh; <sup>ϑ</sup> Dk <sup>þ</sup> <sup>r</sup> � rh; <sup>ϕ</sup>

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

<sup>D</sup><sup>k</sup> þ s � sh; ϕ<sup>x</sup> 

�ð Þ <sup>ν</sup> <sup>þ</sup> <sup>i</sup><sup>η</sup> ð Þ <sup>r</sup> � rh; <sup>ϑ</sup> Dk � <sup>n</sup>^:ð Þ <sup>u</sup> � uh <sup>∗</sup> ð Þ ;<sup>φ</sup> <sup>∂</sup>Dk <sup>¼</sup> <sup>0</sup>:

For the error estimate, we define special projections P� and P<sup>þ</sup> into V<sup>k</sup>

<sup>P</sup><sup>þ</sup> ð Þ <sup>u</sup> � <sup>u</sup>; <sup>v</sup> Dk <sup>¼</sup> <sup>0</sup>, <sup>∀</sup>v<sup>∈</sup> <sup>P</sup><sup>k</sup>

<sup>P</sup>� ð Þ <sup>u</sup> � <sup>u</sup>; <sup>v</sup> <sup>D</sup><sup>k</sup> <sup>¼</sup> <sup>0</sup>, <sup>∀</sup><sup>v</sup> <sup>∈</sup>P<sup>k</sup>�<sup>1</sup>

occurrence) depending solely on u and its derivatives but not of h.

3.2. Error estimates

Dk

Denoting

[21], that

<sup>L</sup>2ð Þ <sup>Ω</sup> <sup>þ</sup> ð Þ sh;sh <sup>L</sup>2ð Þ <sup>Ω</sup> <sup>≤</sup> <sup>γ</sup>∥uh∥<sup>2</sup>

ut ð Þ ; ϑ Dk � ð Þ ν þ iη ð Þ r; ϑ Dk þ ð Þ κ þ iζ ð Þ u; ϑ Dk � γð Þ u; ϑ Dk ¼ 0,

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

<sup>π</sup> <sup>¼</sup> <sup>P</sup>�<sup>u</sup> � uh, <sup>π</sup><sup>e</sup> <sup>¼</sup> <sup>P</sup>�<sup>u</sup> � u, <sup>ε</sup> <sup>¼</sup> <sup>P</sup>þ<sup>r</sup> � rh, <sup>ε</sup><sup>e</sup> <sup>¼</sup> <sup>P</sup>þ<sup>r</sup> � r,

For the abovementioned special projections, we have, by the standard approximation theory

<sup>∥</sup>PþuðÞ�: <sup>u</sup>ð Þ: <sup>∥</sup>L2ð Þ <sup>Ω</sup><sup>h</sup> <sup>≤</sup> ChNþ<sup>1</sup>

<sup>∥</sup>P�uðÞ�: <sup>u</sup>ð Þ: <sup>∥</sup>L2ð Þ <sup>Ω</sup><sup>h</sup> <sup>≤</sup> ChNþ<sup>1</sup>

where here and below C is a positive constant (which may have a different value in each

Dk þ u � uh;φ<sup>x</sup> 

<sup>D</sup><sup>k</sup> <sup>þ</sup> ð Þ <sup>s</sup> � sh;<sup>φ</sup> <sup>D</sup><sup>k</sup> � <sup>n</sup>^:ð Þ <sup>s</sup> � sh <sup>∗</sup>

<sup>N</sup> Dk , <sup>P</sup>þu xk�<sup>1</sup>

<sup>N</sup> <sup>D</sup><sup>k</sup> , <sup>P</sup>�u xkþ<sup>1</sup>

<sup>τ</sup> <sup>¼</sup> <sup>P</sup>þ<sup>s</sup> � sh, <sup>τ</sup><sup>e</sup> <sup>¼</sup> <sup>P</sup>þ<sup>s</sup> � <sup>s</sup>: (22)

<sup>∂</sup>D<sup>k</sup> ,

Dk <sup>þ</sup> <sup>n</sup>^:s<sup>∗</sup>; <sup>ϕ</sup>

Dk <sup>þ</sup> <sup>n</sup>^:u<sup>∗</sup> ð Þ ;<sup>φ</sup> <sup>∂</sup>Dk :

�2γ<sup>T</sup>∥u0ð Þ<sup>x</sup> <sup>∥</sup>Ω: □

� ð Þ ν þ iη Δu þ ð Þ κ þ iζ u � γu ¼ 0: (18)

Local Discontinuous Galerkin Method for Nonlinear Ginzburg-Landau Equation

Dk þ ð Þ κ þ iζ ð Þ u � uh; ϑ Dk

∂D<sup>k</sup>

2 ,

> 2 :

, (23)

<sup>h</sup>. For all the elements,

; ϕ

2 <sup>¼</sup> u xk�<sup>1</sup>

2 <sup>¼</sup> u xkþ<sup>1</sup>

,

<sup>Ω</sup>: (17)

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

(19)

121

(20)

(21)

where n^ is simply a scalar and takes the value of +1 and �1 at the right and the left interface, respectively.
