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<sup>2</sup> stability). The solution given by the LDG method defined by (13) satisfies

∥uhð Þ x; T ∥<sup>Ω</sup> ≤ e �2γ<sup>T</sup>∥u0ð Þ<sup>x</sup> <sup>∥</sup><sup>Ω</sup>

for any T > 0.

Proof. Set <sup>ϑ</sup>; <sup>ϕ</sup>;<sup>φ</sup> <sup>¼</sup> ð Þ uh; <sup>ν</sup>uh; <sup>ν</sup>sh in (13) and consider the integration by parts formula u; <sup>∂</sup><sup>r</sup> ∂x Dk <sup>þ</sup> <sup>r</sup>; <sup>∂</sup><sup>u</sup> ∂x Dk ¼ ½ur� x <sup>k</sup>þ<sup>1</sup> 2 x <sup>k</sup>�<sup>1</sup> , we get

$$\begin{split} & \left( (\boldsymbol{u}\_{h})\_{t}, \boldsymbol{u}\_{h} \right)\_{D^{k}} + (\boldsymbol{s}\_{h}, \boldsymbol{s}\_{h})\_{D^{k}} \\ &= -\nu (\boldsymbol{r}\_{h}, \boldsymbol{u}\_{h})\_{D^{k}} + (\boldsymbol{\nu} + \boldsymbol{i}\boldsymbol{\eta}) (\boldsymbol{r}\_{h}, \boldsymbol{u}\_{h})\_{D^{k}} - (\boldsymbol{\kappa} + \boldsymbol{i}\boldsymbol{\zeta}) \left( |\boldsymbol{u}\_{h}|^{2} \boldsymbol{u}\_{h}, \boldsymbol{u}\_{h} \right)\_{D^{k}} \\ &+ \gamma (\boldsymbol{u}\_{h}, \boldsymbol{u}\_{h})\_{D^{k}} + \nu \left( \hat{\boldsymbol{n}}, \boldsymbol{s}\_{h}^{\*}, \boldsymbol{u}\_{h} \right)\_{\partial D^{k}} + \nu \left( \hat{\boldsymbol{n}}, \boldsymbol{u}\_{h}^{\*}, \boldsymbol{s}\_{h} \right)\_{\partial D^{k}} - \nu (\hat{\boldsymbol{n}}, \boldsymbol{s}\_{h}, \boldsymbol{u}\_{h})\_{\partial D^{k}}. \end{split} \tag{14}$$

Taking the real part of the resulting equation, we obtain

2

$$\begin{split} \left( (\boldsymbol{\mu}\_{\boldsymbol{h}})\_{t}, \boldsymbol{\mu}\_{\boldsymbol{h}} \right)\_{D^{k}} + (\mathbf{s}\_{\boldsymbol{h}}, \mathbf{s}\_{\boldsymbol{h}})\_{D^{k}} &= -\kappa \Big( |\boldsymbol{\mu}\_{\boldsymbol{h}}|^{2} \boldsymbol{\mu}\_{\boldsymbol{h}}, \boldsymbol{\mu}\_{\boldsymbol{h}} \Big)\_{D^{k}} + \gamma (\boldsymbol{\mu}\_{\boldsymbol{h}}, \boldsymbol{\mu}\_{\boldsymbol{h}})\_{D^{k}} \\ &+ \nu \Big( \hat{\boldsymbol{n}}, \mathbf{s}\_{\boldsymbol{h}}^{\*}, \boldsymbol{\mu}\_{\boldsymbol{h}} \Big)\_{\partial D^{k}} + \nu \Big( \hat{\boldsymbol{n}}. \boldsymbol{\mu}\_{\boldsymbol{h}}^{\*}, \mathbf{s}\_{\boldsymbol{h}} \Big)\_{\partial D^{k}} - \nu (\hat{\boldsymbol{n}}. \boldsymbol{s}\_{\boldsymbol{h}}, \boldsymbol{\mu}\_{\boldsymbol{h}})\_{\partial D^{k}}. \end{split} \tag{15}$$

Removing the positive term κ j j uh 2 uh; uh Dk , we obtain

$$\|\left(\left(\boldsymbol{\mu}\_{\boldsymbol{h}}\right)\_{\boldsymbol{l}},\boldsymbol{\mu}\_{\boldsymbol{h}}\right)\_{\boldsymbol{D}^{\boldsymbol{k}}}+\left(\boldsymbol{s}\_{\boldsymbol{h}},\boldsymbol{s}\_{\boldsymbol{h}}\right)\_{\boldsymbol{D}^{\boldsymbol{k}}}\leq\nu\|\boldsymbol{\mu}\_{\boldsymbol{h}}\|\_{\boldsymbol{L}^{2}\left(\boldsymbol{D}^{\boldsymbol{k}}\right)}^{2}+\nu\left(\left\|\boldsymbol{\cdot}.\boldsymbol{s}\_{\boldsymbol{h}}^{\*},\boldsymbol{\mu}\_{\boldsymbol{h}}\right\|\_{\boldsymbol{\partial}\boldsymbol{D}^{\boldsymbol{k}}}+\nu\left(\left\|\boldsymbol{\cdot}.\boldsymbol{u}\_{\boldsymbol{h}}^{\*},\boldsymbol{s}\_{\boldsymbol{h}}\right\|\_{\boldsymbol{\partial}\boldsymbol{D}^{\boldsymbol{k}}}-\nu\left(\left\|\boldsymbol{\cdot}.\boldsymbol{s}\_{\boldsymbol{h}},\boldsymbol{\mu}\_{\boldsymbol{h}}\right\|\_{\boldsymbol{\partial}\boldsymbol{D}^{\boldsymbol{k}}}\right.\right.\right.\tag{16}$$

Summing over all elements (16), we easily obtain

Local Discontinuous Galerkin Method for Nonlinear Ginzburg-Landau Equation http://dx.doi.org/10.5772/intechopen.75300 121

$$\|\left(\left(\mu\_{\hbar}\right)\_{t},\mu\_{\hbar}\right)\_{L^{2}\left(\Omega\right)} + \left(s\_{\hbar},s\_{\hbar}\right)\_{L^{2}\left(\Omega\right)} \lesssim \gamma \|\mu\_{\hbar}\|\_{\Omega}^{2}.\tag{17}$$

Employing Gronwall's inequality, we obtain

$$\|\mu\_h(\mathfrak{x}, T)\|\_{\Omega} \le e^{-2\gamma T} \|\mu\_0(\mathfrak{x})\|\_{\Omega}. \quad \Box$$

#### 3.2. Error estimates

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

120 Differential Equations - Theory and Current Research

rh; ϕ

respectively.

Landau problem.

for any T > 0.

Dk <sup>þ</sup> <sup>r</sup>; <sup>∂</sup><sup>u</sup> ∂x 

> ð Þ uh <sup>t</sup> ; uh

ð Þ uh <sup>t</sup> ; uh 

Removing the positive term κ j j uh

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

Summing over all elements (16), we easily obtain

Dk ¼ ½ur� x <sup>k</sup>þ<sup>1</sup> 2 x <sup>k</sup>�<sup>1</sup> 2

ð Þ uh <sup>t</sup> ; uh  , we get

<sup>þ</sup>γð Þ uh; uh Dk <sup>þ</sup> <sup>ν</sup> <sup>n</sup>^:s<sup>∗</sup>

Taking the real part of the resulting equation, we obtain

Dk þ ð Þ sh;sh Dk

Dk þ ð Þ sh;sh Dk ¼ � κ j j uh

2 uh; uh 

u; <sup>∂</sup><sup>r</sup> ∂x 

3.1. Stability analysis

Dk ¼ � sh; ϕ<sup>x</sup> 

ð Þ sh; φ Dk ¼ � uh;φ<sup>x</sup>

3. Stability and error estimates

<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>n</sup>^:s<sup>∗</sup> <sup>h</sup>; <sup>ϕ</sup>

<sup>D</sup><sup>k</sup> <sup>þ</sup> <sup>n</sup>^:u<sup>∗</sup> <sup>h</sup>;<sup>φ</sup>

2 uh; ϑ 

<sup>∂</sup>Dk ,

<sup>∂</sup>Dk :

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

In this section, we discuss stability and accuracy of the proposed scheme, for the Ginzburg-

Proof. Set <sup>ϑ</sup>; <sup>ϕ</sup>;<sup>φ</sup> <sup>¼</sup> ð Þ uh; <sup>ν</sup>uh; <sup>ν</sup>sh in (13) and consider the integration by parts formula

<sup>∂</sup>Dk <sup>þ</sup> <sup>ν</sup> <sup>n</sup>^:u<sup>∗</sup>

, we obtain

∗ <sup>h</sup>; uh 

2 uh; uh 

<sup>h</sup>;sh 

<sup>∂</sup>D<sup>k</sup> <sup>þ</sup> <sup>ν</sup> <sup>n</sup>^:u<sup>∗</sup>

<sup>D</sup><sup>k</sup> <sup>þ</sup> <sup>γ</sup>ð Þ uh; uh Dk

<sup>∂</sup>Dk <sup>þ</sup> <sup>ν</sup> <sup>n</sup>^:u<sup>∗</sup>

<sup>h</sup>;sh 

> <sup>h</sup>;sh

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

2 uh; uh 

<sup>∂</sup>Dk � νð Þ n^:sh; uh <sup>∂</sup>Dk :

Dk

<sup>∂</sup>D<sup>k</sup> � νð Þ n^:sh; uh <sup>∂</sup>D<sup>k</sup> :

<sup>∂</sup>Dk � νð Þ n^:sh; uh <sup>∂</sup>Dk : (16)

In order to carry out the analysis of the LDG scheme, we have the following results. Theorem 3.1. (L<sup>2</sup> stability). The solution given by the LDG method defined by (13) satisfies

∥uhð Þ x; T ∥<sup>Ω</sup> ≤ e

¼ �νð Þ rh; uh <sup>D</sup><sup>k</sup> þ ð Þ ν þ iη ð Þ rh; uh <sup>D</sup><sup>k</sup> � ð Þ κ þ iζ j j uh

<sup>h</sup>; uh 

> þ ν n^:s ∗ <sup>h</sup>; uh

<sup>L</sup><sup>2</sup> Dk ð Þ <sup>þ</sup> <sup>ν</sup> <sup>n</sup>^:<sup>s</sup>

Dk

<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>,

(13)

(14)

(15)

We consider the linear Ginzburg-Landau equation

$$\frac{\partial u}{\partial t} - (\nu + i\eta)\Delta u + (\kappa + i\zeta)u - \gamma u = 0. \tag{18}$$

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

$$\begin{aligned} & (\boldsymbol{u}, \boldsymbol{\uprho})\_{D^{k}} - (\boldsymbol{\upnu} + i\boldsymbol{\upeta})(\boldsymbol{r}, \boldsymbol{\uprho})\_{D^{k}} + (\boldsymbol{\upkappa} + i\boldsymbol{\upzeta})(\boldsymbol{u}, \boldsymbol{\uprho})\_{D^{k}} - \boldsymbol{\uprho}(\boldsymbol{u}, \boldsymbol{\uprho})\_{D^{k}} = \boldsymbol{0}, \\ & (\boldsymbol{r}, \boldsymbol{\uprho})\_{D^{\ell}} = -\left(\boldsymbol{s}, \boldsymbol{\uprho}\right)\_{D^{\ell}} + \left(\hat{\boldsymbol{n}}, \boldsymbol{s}^{\*}, \boldsymbol{\uprho}\right)\_{\partial D^{k}} \\ & (\boldsymbol{s}, \boldsymbol{\uprho})\_{D^{\ell}} = -\left(\boldsymbol{u}, \boldsymbol{\uprho}\_{\boldsymbol{x}}\right)\_{D^{\ell}} + \left(\hat{\boldsymbol{n}}.\boldsymbol{u}^{\*}, \boldsymbol{\uprho}\right)\_{\partial D^{\ell}}. \end{aligned} \tag{19}$$

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

$$\begin{cases} \left( (\boldsymbol{\mu} - \boldsymbol{\mu}\_{h})\_{t}, \mathfrak{d} \right)\_{D^{k}} + \left( \boldsymbol{s} - \boldsymbol{s}\_{h}, \boldsymbol{\phi}\_{x} \right)\_{D^{k}} + \left( \boldsymbol{\mu} - \boldsymbol{\mu}\_{h}, \boldsymbol{\phi}\_{x} \right)\_{D^{k}} + \left( \boldsymbol{\kappa} + \boldsymbol{i} \zeta \right) (\boldsymbol{\mu} - \boldsymbol{\mu}\_{h}, \boldsymbol{\mathfrak{d}})\_{D^{k}} \\ \quad - \gamma (\boldsymbol{\mu} - \boldsymbol{\mu}\_{h}, \boldsymbol{\mathfrak{d}})\_{D^{k}} + \left( r - r\_{h}, \boldsymbol{\phi} \right)\_{D^{k}} + \left( \boldsymbol{s} - \boldsymbol{s}\_{h}, \boldsymbol{\boldsymbol{\phi}} \right)\_{D^{k}} - \left( \boldsymbol{\hat{n}}. \left( \boldsymbol{s} - \boldsymbol{s}\_{h} \right)^{\*}, \boldsymbol{\phi} \right)\_{\partial D^{k}} \\ \quad - \left( \boldsymbol{\nu} + \boldsymbol{i} \boldsymbol{\eta} \right) (r - r\_{h}, \boldsymbol{\mathfrak{d}})\_{D^{k}} - \left( \boldsymbol{\hat{n}}. \left( \boldsymbol{\mu} - \boldsymbol{\mu}\_{h} \right)^{\*}, \boldsymbol{\uprho} \right)\_{\partial D^{k}} = \boldsymbol{0}. \end{cases} \tag{20}$$

For the error estimate, we define special projections P� and P<sup>þ</sup> into V<sup>k</sup> <sup>h</sup>. For all the elements, Dk , k ¼ 1, 2, …, K are defined to satisfy

$$\begin{split} (\mathcal{P}^+ u - u, \boldsymbol{\upsilon})\_{D^k} &= 0, \quad \forall \boldsymbol{\upsilon} \in \mathbb{P}\_N^k(\boldsymbol{D}^k), \quad \mathcal{P}^+ u \left(\mathbf{x}\_{k - \frac{1}{2}}\right) = u \left(\mathbf{x}\_{k - \frac{1}{2}}\right), \\ (\mathcal{P}^- u - u, \boldsymbol{\upsilon})\_{D^k} &= 0, \quad \forall \boldsymbol{\upsilon} \in \mathbb{P}\_N^{k - 1}(\boldsymbol{D}^k), \quad \mathcal{P}^- u \left(\mathbf{x}\_{k + \frac{1}{2}}\right) = u \left(\mathbf{x}\_{k + \frac{1}{2}}\right). \end{split} \tag{21}$$

Denoting

$$\begin{aligned} \pi &= \mathcal{P}^- \boldsymbol{u} - \boldsymbol{u}\_{\boldsymbol{h}\boldsymbol{\nu}} & \quad \pi^\varepsilon = \mathcal{P}^- \boldsymbol{u} - \boldsymbol{u}, \quad \boldsymbol{\varepsilon} = \mathcal{P}^+ \boldsymbol{r} - \boldsymbol{r}\_{\boldsymbol{h}\boldsymbol{\nu}} & \quad \boldsymbol{\varepsilon}^\varepsilon = \mathcal{P}^+ \boldsymbol{r} - \boldsymbol{r}, \\ \pi &= \mathcal{P}^+ \boldsymbol{s} - \boldsymbol{s}\_{\boldsymbol{h}\boldsymbol{\nu}} & \quad \boldsymbol{\tau}^\varepsilon = \mathcal{P}^+ \boldsymbol{s} - \boldsymbol{s}. \end{aligned} \tag{22}$$

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

$$\begin{aligned} \|\mathcal{P}^+ u(.) - u(.)\|\_{L^2(\Omega\_b)} &\le \mathcal{C} h^{N+1}, \\ \|\mathcal{P}^- u(.) - u(.)\|\_{L^2(\Omega\_b)} &\le \mathcal{C} h^{N+1}, \end{aligned} \tag{23}$$

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:

$$\|\|\mu(.,t) - \mu\_h(.,t)\|\|\_{L^2(\Omega\_h)} \le Ch^{N+1},\tag{24}$$

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

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

<sup>π</sup> � <sup>π</sup><sup>e</sup> ð Þ<sup>t</sup> ; ϑ � � Dk <sup>þ</sup> <sup>τ</sup> � <sup>τ</sup><sup>e</sup> ; ϕ<sup>x</sup> � � Dk <sup>þ</sup> <sup>π</sup> � <sup>π</sup><sup>e</sup> ;φ<sup>x</sup> � � <sup>D</sup><sup>k</sup> <sup>þ</sup> ð Þ <sup>κ</sup> <sup>þ</sup> <sup>i</sup><sup>ζ</sup> <sup>π</sup> � <sup>π</sup><sup>e</sup> ð Þ ; <sup>ϑ</sup> <sup>D</sup><sup>k</sup> � γ π � <sup>π</sup><sup>e</sup> ð Þ ; <sup>ϑ</sup> Dk <sup>þ</sup> <sup>ε</sup> � <sup>ε</sup><sup>e</sup> ; ϕ � � Dk <sup>þ</sup> <sup>τ</sup> � <sup>τ</sup><sup>e</sup> ð Þ ;<sup>φ</sup> Dk <sup>þ</sup> <sup>ϕ</sup> � <sup>ϕ</sup><sup>e</sup> ; β � � <sup>D</sup><sup>k</sup> � <sup>n</sup>^: <sup>τ</sup> � <sup>τ</sup><sup>e</sup> ð Þ<sup>∗</sup> ; ϕ � � <sup>∂</sup>D<sup>k</sup> � ð Þ ν þ iη � <sup>ε</sup> � <sup>ε</sup><sup>e</sup> ð Þ ; <sup>ϑ</sup> <sup>D</sup><sup>k</sup> � <sup>n</sup>^: <sup>π</sup> � <sup>π</sup><sup>e</sup> ð Þ<sup>∗</sup> ð Þ ;<sup>φ</sup> <sup>∂</sup>Dk <sup>¼</sup> <sup>0</sup>:

Taking the real part of the resulting equation, we obtain

$$\begin{cases} \left(\pi-\pi^{\varepsilon}\right)\_{t},\mathfrak{d}\right\}\_{D^{k}}+\left(\pi-\pi^{\varepsilon},\phi\_{\mathfrak{x}}\right)\_{D^{k}}+\left(\pi-\pi^{\varepsilon},\varphi\_{\mathfrak{x}}\right)\_{D^{k}}+\kappa\left(\pi-\pi^{\varepsilon},\mathfrak{d}\right)\_{D^{k}}\\ \quad -\gamma\left(\pi-\pi^{\varepsilon},\mathfrak{d}\right)\_{D^{k}}+\left(\varepsilon-\varepsilon^{\varepsilon},\phi\right)\_{D^{k}}+\left(\pi-\pi^{\varepsilon},\varphi\right)\_{D^{k}}-\left(\mathring{\mathfrak{n}}.\left(\pi-\pi^{\varepsilon}\right)^{\*},\phi\right)\_{\partial D^{k}}\\ \quad -\nu\left(\varepsilon-\varepsilon^{\varepsilon},\mathfrak{d}\right)\_{D^{k}}-\left(\mathring{\mathfrak{n}}.\left(\pi-\pi^{\varepsilon}\right)^{\*},\varphi\right)\_{\partial D^{k}}=0. \end{cases} \tag{26}$$

We take the test functions

$$
\delta \Phi = \pi, \quad \Phi = \nu \pi, \quad \varphi = \nu \pi,\tag{27}
$$

(25)

III ¼ γ πð Þ ; π <sup>Ω</sup> � κ πð Þ ; π <sup>Ω</sup>: (33)

Local Discontinuous Galerkin Method for Nonlinear Ginzburg-Landau Equation

<sup>L</sup>2ð Þ <sup>Ω</sup> <sup>þ</sup> Ch<sup>2</sup>Nþ<sup>2</sup>

<sup>L</sup>2ð Þ <sup>Ω</sup> <sup>þ</sup> <sup>c</sup>2∥τ∥<sup>2</sup>

<sup>L</sup>2ð Þ <sup>Ω</sup> <sup>þ</sup> Ch<sup>2</sup>Nþ<sup>2</sup>

I ¼ 0: (34)

<sup>L</sup>2ð Þ <sup>Ω</sup> : (36)

<sup>L</sup>2ð Þ <sup>Ω</sup> <sup>þ</sup> Ch<sup>2</sup>Nþ<sup>2</sup>

<sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>F</sup>ð Þ <sup>u</sup>h; <sup>t</sup> , (39)

: (35)

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

123

, (37)

(40)

: (38)

Using the definitions of the projections P, S (21) in (31), we get

II ≤ c1∥π∥<sup>2</sup>

<sup>π</sup><sup>t</sup> ð Þ ; <sup>π</sup> <sup>Ω</sup> <sup>þ</sup> ν τð Þ ; <sup>τ</sup> <sup>Ω</sup> <sup>≤</sup> <sup>c</sup>1∥π∥<sup>2</sup>

<sup>π</sup><sup>t</sup> ð Þ ; <sup>π</sup> <sup>Ω</sup> <sup>≤</sup> <sup>c</sup>1∥π∥<sup>2</sup>

∂u<sup>h</sup>

h ; t <sup>n</sup> ,

> <sup>h</sup> þ 1 2 Δtk<sup>1</sup> ; t n þ 1 2 Δt

> <sup>h</sup> þ 1 2 Δtk<sup>2</sup> ; t n þ 1 2 Δt

<sup>h</sup> <sup>þ</sup> <sup>Δ</sup>tk<sup>3</sup> ;t <sup>n</sup> <sup>þ</sup> <sup>Δ</sup><sup>t</sup> ,

<sup>k</sup><sup>1</sup> <sup>¼</sup> <sup>F</sup> <sup>u</sup><sup>n</sup>

<sup>k</sup><sup>2</sup> <sup>¼</sup> <sup>F</sup> <sup>u</sup><sup>n</sup>

<sup>k</sup><sup>3</sup> <sup>¼</sup> <sup>F</sup> <sup>u</sup><sup>n</sup>

<sup>k</sup><sup>4</sup> <sup>¼</sup> <sup>F</sup> <sup>u</sup><sup>n</sup>

u<sup>n</sup>þ<sup>1</sup> <sup>h</sup> <sup>¼</sup> <sup>u</sup><sup>n</sup> <sup>h</sup> þ 1

provided c<sup>2</sup> is sufficiently small such that c<sup>2</sup> ≤ ν, we obtain that

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

4. Numerical examples

stability

RK method (ERK)

and

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

<sup>L</sup>2ð Þ <sup>Ω</sup> <sup>þ</sup> <sup>c</sup>2∥τ∥<sup>2</sup>

III ≤ c1∥π∥<sup>2</sup>

From the Gronwall's lemma and standard approximation theory, the desired result follows. ⃞.

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

where u<sup>h</sup> is the vector of unknowns, we can use the standard fourth-order four-stage explicit

,

,

<sup>6</sup> <sup>k</sup><sup>1</sup> <sup>þ</sup> <sup>2</sup>k<sup>2</sup> <sup>þ</sup> <sup>2</sup>k<sup>3</sup> <sup>þ</sup> <sup>k</sup><sup>4</sup> ,

we obtain

$$\begin{cases} \left( (\pi - \pi^{\varepsilon})\_{t}, \pi \right)\_{D^{k}} + \nu (\pi - \pi^{\varepsilon}, \pi\_{\mathfrak{x}})\_{D^{k}} + \nu (\pi - \pi^{\varepsilon}, \pi\_{\mathfrak{x}})\_{D^{k}} \\ + \kappa (\pi - \pi^{\varepsilon}, \pi)\_{D^{k}} - \gamma (\pi - \pi^{\varepsilon}, \pi)\_{D^{k}} + \nu (\varepsilon - \varepsilon^{\varepsilon}, \pi)\_{D^{k}} \\ + \nu (\pi - \pi^{\varepsilon}, \pi)\_{D^{k}} - \nu (\hat{\mathfrak{n}}. \left( \pi - \pi^{\varepsilon} \right)^{\*}, \pi \rangle\_{\partial D^{k}} - \nu (\varepsilon - \varepsilon^{\varepsilon}, \pi)\_{D^{k}} - \nu (\hat{\mathfrak{n}}. \left( \pi - \pi^{\varepsilon} \right)^{\*}, \pi)\_{\partial D^{k}} = 0. \end{cases} \tag{28}$$

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

$$\begin{split} \left(\pi\_{t},\pi\right)\_{\Omega} + \nu(\pi,\pi)\_{\Omega} &= \nu(\pi^{\epsilon},\pi\_{\mathfrak{x}})\_{\Omega} + \nu(\pi^{\epsilon},\pi\_{\mathfrak{x}})\_{\Omega} + (\pi^{\epsilon}\_{t},\pi)\_{\Omega} - \gamma(\pi^{\epsilon},\pi)\_{\Omega} + \kappa(\pi^{\epsilon},\pi)\_{\Omega} \\ + \nu(\pi^{\epsilon},\pi)\_{\Omega} + \gamma(\pi,\pi)\_{\Omega} &- \nu\sum\_{k=1}^{K} \left(\hat{n}.\left(\pi^{\epsilon}\right)^{\*},\pi\right)\_{\partial\mathcal{D}^{k}} - \nu\sum\_{k=1}^{K} \left(\hat{n}.\left(\pi^{\epsilon}\right)^{\*},\pi\right)\_{\partial\mathcal{D}^{k}} \end{split} \tag{29}$$

we can rewrite (29) as

$$(\pi\_t, \pi)\_{\Omega} + \nu(\pi, \pi)\_{\Omega} = I + II + III,\tag{30}$$

where

$$I = \nu(\pi^\epsilon, \pi\_\mathbf{x})\_{\Omega} + \nu(\pi^\epsilon, \pi\_\mathbf{x})\_{\Omega'} \tag{31}$$

$$\begin{split} II &= \left(\pi\_t^\epsilon, \pi\right)\_\Omega - \gamma \left(\pi^\epsilon, \pi\right)\_\Omega + \kappa \left(\pi^\epsilon, \pi\right)\_\Omega + \nu \left(\pi^\epsilon, \pi\right)\_\Omega \\ & \quad - \nu \sum\_{k=1}^K \left(\hat{n}. \left(\pi^\epsilon\right)^\*, \pi\right)\_{\partial\mathcal{D}^k} - \nu \sum\_{k=1}^K \left(\hat{n}. \left(\pi^\epsilon\right)^\*, \pi\right)\_{\partial\mathcal{D}^k} \end{split} \tag{32}$$

Local Discontinuous Galerkin Method for Nonlinear Ginzburg-Landau Equation http://dx.doi.org/10.5772/intechopen.75300 123

$$
\Pi = \gamma(\pi, \pi)\_{\Omega} - \kappa(\pi, \pi)\_{\Omega}. \tag{33}
$$

Using the definitions of the projections P, S (21) in (31), we get

$$I = 0.\tag{34}$$

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

$$\|\Pi \le c\_1 \|\|\pi\|\|\_{L^2(\Omega)}^2 + c\_2 \|\|\pi\|\|\_{L^2(\Omega)}^2 + C h^{2N+2}.\tag{35}$$

and

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:

<sup>∥</sup>uð Þ� :; <sup>t</sup> uhð Þ :; <sup>t</sup> <sup>∥</sup>L2ð Þ <sup>Ω</sup><sup>h</sup> <sup>≤</sup>ChNþ<sup>1</sup>

;φ<sup>x</sup> � �

Dk <sup>þ</sup> <sup>π</sup> � <sup>π</sup><sup>e</sup>

<sup>D</sup><sup>k</sup> � <sup>n</sup>^: <sup>τ</sup> � <sup>τ</sup><sup>e</sup> ð Þ<sup>∗</sup> ; ϕ � �

;φ<sup>x</sup> � �

Dk <sup>þ</sup> <sup>τ</sup> � <sup>τ</sup><sup>e</sup> ð Þ ;<sup>φ</sup> Dk � <sup>n</sup>^: <sup>τ</sup> � <sup>τ</sup><sup>e</sup> ð Þ<sup>∗</sup>

ϑ ¼ π, ϕ ¼ νπ, φ ¼ ντ, (27)

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

Dk <sup>þ</sup> <sup>π</sup> � <sup>π</sup><sup>e</sup>

; β � �

; ϕ<sup>x</sup> � �

; ϕ � �

<sup>D</sup><sup>k</sup> <sup>þ</sup> ν τ � <sup>τ</sup><sup>e</sup> ð Þ ; <sup>π</sup><sup>x</sup> Dk <sup>þ</sup> ν π � <sup>π</sup><sup>e</sup> ð Þ ; <sup>τ</sup><sup>x</sup> Dk

<sup>þ</sup>ν τ � <sup>τ</sup><sup>e</sup> ð Þ ; <sup>τ</sup> <sup>D</sup><sup>k</sup> � <sup>ν</sup> <sup>n</sup>^: <sup>τ</sup> � <sup>τ</sup><sup>e</sup> ð Þ<sup>∗</sup> ð Þ ; <sup>π</sup> <sup>∂</sup>D<sup>k</sup> � ν ε � <sup>ε</sup><sup>e</sup> ð Þ ; <sup>π</sup> <sup>D</sup><sup>k</sup> � <sup>ν</sup> <sup>n</sup>^: <sup>π</sup> � <sup>π</sup><sup>e</sup> ð Þ<sup>∗</sup> ð Þ ; <sup>τ</sup> <sup>∂</sup>Dk <sup>¼</sup> <sup>0</sup>:

X K

<sup>n</sup>^: <sup>π</sup><sup>e</sup> ð Þ<sup>∗</sup> ð Þ ; <sup>τ</sup> <sup>∂</sup>Dk � <sup>ν</sup>

<sup>Ω</sup> � γ π<sup>e</sup> ð Þ ; <sup>π</sup> <sup>Ω</sup> <sup>þ</sup> κ π<sup>e</sup> ð Þ ; <sup>π</sup> <sup>Ω</sup> <sup>þ</sup> ν τ<sup>e</sup> ð Þ ; <sup>τ</sup> <sup>Ω</sup>

X K

k¼1

k¼1

<sup>n</sup>^: <sup>π</sup><sup>e</sup> ð Þ<sup>∗</sup> ð Þ ; <sup>τ</sup> <sup>∂</sup>D<sup>k</sup> � <sup>ν</sup>

<sup>þ</sup>κ π � <sup>π</sup><sup>e</sup> ð Þ ; <sup>π</sup> Dk � γ π � <sup>π</sup><sup>e</sup> ð Þ ; <sup>π</sup> Dk <sup>þ</sup> ν ε � <sup>ε</sup><sup>e</sup> ð Þ ; <sup>π</sup> <sup>D</sup><sup>k</sup>

Summing over k, simplify by integration by parts and (9), we get <sup>π</sup><sup>t</sup> ð Þ ; <sup>π</sup> <sup>Ω</sup> <sup>þ</sup> ν τð Þ ; <sup>τ</sup> <sup>Ω</sup> <sup>¼</sup> ν τ<sup>e</sup> ð Þ ; <sup>π</sup><sup>x</sup> <sup>Ω</sup> <sup>þ</sup> ν π<sup>e</sup> ð Þ ; <sup>τ</sup><sup>x</sup> <sup>Ω</sup> <sup>þ</sup> <sup>π</sup><sup>e</sup>

<sup>þ</sup>ν τ<sup>e</sup> ð Þ ; <sup>τ</sup> <sup>Ω</sup> <sup>þ</sup> γ πð Þ ; <sup>π</sup> <sup>Ω</sup> � κ πð Þ ; <sup>π</sup> <sup>Ω</sup> � <sup>ν</sup>

II <sup>¼</sup> <sup>π</sup><sup>e</sup> <sup>t</sup>; <sup>π</sup> � �

> �ν X K

> > k¼1

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

; ϕ<sup>x</sup> � �

Dk <sup>þ</sup> <sup>τ</sup> � <sup>τ</sup><sup>e</sup> ð Þ ;<sup>φ</sup> Dk <sup>þ</sup> <sup>ϕ</sup> � <sup>ϕ</sup><sup>e</sup>

Taking the real part of the resulting equation, we obtain

�γ π � <sup>π</sup><sup>e</sup> ð Þ ; <sup>ϑ</sup> Dk <sup>þ</sup> <sup>ε</sup> � <sup>ε</sup><sup>e</sup>

Dk <sup>þ</sup> <sup>τ</sup> � <sup>τ</sup><sup>e</sup>

�ν ε � <sup>ε</sup><sup>e</sup> ð Þ ; <sup>ϑ</sup> Dk � <sup>n</sup>^: <sup>π</sup> � <sup>π</sup><sup>e</sup> ð Þ<sup>∗</sup> ð Þ ;<sup>φ</sup> <sup>∂</sup>Dk <sup>¼</sup> <sup>0</sup>:

Dk <sup>þ</sup> <sup>τ</sup> � <sup>τ</sup><sup>e</sup>

122 Differential Equations - Theory and Current Research

� <sup>ε</sup> � <sup>ε</sup><sup>e</sup> ð Þ ; <sup>ϑ</sup> <sup>D</sup><sup>k</sup> � <sup>n</sup>^: <sup>π</sup> � <sup>π</sup><sup>e</sup> ð Þ<sup>∗</sup> ð Þ ;<sup>φ</sup> <sup>∂</sup>Dk <sup>¼</sup> <sup>0</sup>:

<sup>π</sup> � <sup>π</sup><sup>e</sup> ð Þ<sup>t</sup> ; ϑ � �

<sup>π</sup> � <sup>π</sup><sup>e</sup> ð Þ<sup>t</sup> ; ϑ � �

We take the test functions

<sup>π</sup> � <sup>π</sup><sup>e</sup> ð Þ<sup>t</sup> ; π � �

we can rewrite (29) as

where

we obtain

<sup>þ</sup> <sup>ε</sup> � <sup>ε</sup><sup>e</sup> ; ϕ � � , (24)

(25)

(26)

(28)

<sup>D</sup><sup>k</sup> <sup>þ</sup> ð Þ <sup>κ</sup> <sup>þ</sup> <sup>i</sup><sup>ζ</sup> <sup>π</sup> � <sup>π</sup><sup>e</sup> ð Þ ; <sup>ϑ</sup> <sup>D</sup><sup>k</sup> � γ π � <sup>π</sup><sup>e</sup> ð Þ ; <sup>ϑ</sup> Dk

Dk <sup>þ</sup> κ π � <sup>π</sup><sup>e</sup> ð Þ ; <sup>ϑ</sup> <sup>D</sup><sup>k</sup>

; ϕ � �

<sup>t</sup> ð Þ ; <sup>π</sup> <sup>Ω</sup> � γ π<sup>e</sup> ð Þ ; <sup>π</sup> <sup>Ω</sup> <sup>þ</sup> κ π<sup>e</sup> ð Þ ; <sup>π</sup> <sup>Ω</sup>

<sup>n</sup>^: <sup>τ</sup><sup>e</sup> ð Þ<sup>∗</sup> ð Þ ; <sup>π</sup> <sup>∂</sup>Dk , (29)

<sup>n</sup>^: <sup>τ</sup><sup>e</sup> ð Þ<sup>∗</sup> ð Þ ; <sup>π</sup> <sup>∂</sup>Dk , (32)

X K

k¼1

π<sup>t</sup> ð Þ ; π <sup>Ω</sup> þ ν τð Þ ; τ <sup>Ω</sup> ¼ I þ II þ III, (30)

<sup>I</sup> <sup>¼</sup> ν τ<sup>e</sup> ð Þ ; <sup>π</sup><sup>x</sup> <sup>Ω</sup> <sup>þ</sup> ν π<sup>e</sup> ð Þ ; <sup>τ</sup><sup>x</sup> <sup>Ω</sup>, (31)

∂Dk

<sup>∂</sup>D<sup>k</sup> � ð Þ ν þ iη

$$III \le c\_1 \left\| \pi \right\|\_{L^2(\Omega)}^2. \tag{36}$$

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

$$\|\pi(\pi\_t, \pi)\_\Omega + \nu(\tau, \tau)\_\Omega \le c\_1 \|\|\pi\|\|\_{L^2(\Omega)}^2 + c\_2 \|\|\tau\|\|\_{L^2(\Omega)}^2 + C\hbar^{2N+2},\tag{37}$$

provided c<sup>2</sup> is sufficiently small such that c<sup>2</sup> ≤ ν, we obtain that

$$\|(\pi\_t, \pi)\_{\Omega} \le c\_1 \|\pi\|\_{L^2(\Omega)}^2 + \mathcal{C}h^{2N+2}.\tag{38}$$

From the Gronwall's lemma and standard approximation theory, the desired result follows. ⃞.
