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

$$\frac{\partial \mathbf{u}\_h}{\partial t} = \mathcal{F}(\mathbf{u}\_h, t), \tag{39}$$

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

$$\begin{aligned} \mathbf{k}^1 &= \mathcal{F}(\mathbf{u}\_h^u, t^\mu), \\ \mathbf{k}^2 &= \mathcal{F}\left(\mathbf{u}\_h^u + \frac{1}{2}\Delta t \mathbf{k}^1, t^\mu + \frac{1}{2}\Delta t\right), \\ \mathbf{k}^3 &= \mathcal{F}\left(\mathbf{u}\_h^u + \frac{1}{2}\Delta t \mathbf{k}^2, t^\mu + \frac{1}{2}\Delta t\right), \\ \mathbf{k}^4 &= \mathcal{F}\left(\mathbf{u}\_h^u + \Delta t \mathbf{k}^3, t^\mu + \Delta t\right), \\ \mathbf{u}\_h^{n+1} &= \mathbf{u}\_h^n + \frac{1}{6}\left(\mathbf{k}^1 + 2\mathbf{k}^2 + 2\mathbf{k}^3 + \mathbf{k}^4\right), \end{aligned} \tag{40}$$

to advance from u<sup>n</sup> <sup>h</sup> to u<sup>n</sup>þ<sup>1</sup> <sup>h</sup> , separated by the time step, Δt. In our examples, the condition Δt ≤ CΔx<sup>α</sup> min ð Þ 0 < C < 1 is used to ensure stability.

Example 4.1 We consider the following linear Ginzburg-Landau equation

$$\frac{\partial \mu}{\partial t} - (\nu + i\eta)\Delta \mu + (\kappa + i\zeta)\mu = 0, \quad \mathbf{x} \in [-20, 20], \quad t \in (0, 0.5], \quad \mu(\mathbf{x}, 0) = \mu\_0(\mathbf{x}), \tag{41}$$

with

$$\eta = \frac{1}{2}, \kappa = -\frac{\nu \left(3\sqrt{1 + 4\nu^2} - 1\right)}{2(2 + 9\nu^2)}, \zeta = -1, \gamma = 0. \tag{42}$$

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

especially for γ < 0 and the parameter γ dramatically affects the wave shape.

Figure 2. Numerical results for the nonlinear Ginzburg-Landau equation in Example 4.2.

u xð Þ¼ ; <sup>0</sup> <sup>e</sup>�x<sup>2</sup>

with parameters ν ¼ 1, κ ¼ 1, η ¼ 1, ζ ¼ 2, x ∈½ � �10; 10 . We consider cases with N = 2 and K = 40 and solve the equation for several different values of γ. The numerical solution uhð Þ x; t for γ ¼ 2, 1, 0, � 1, � 2 is shown in Figures 2 and 3. The parameter γ will affect the wave shape. From these figures, it is obvious that the solution decays rapidly with time evolution

, (44)

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125

Local Discontinuous Galerkin Method for Nonlinear Ginzburg-Landau Equation

The exact solution u xð Þ¼ ; <sup>t</sup> a xð Þeidln ð Þ� a xð Þ <sup>i</sup>ω<sup>t</sup> where

$$a(\mathbf{x}) = Fsc ch(\mathbf{x}),\\F = \sqrt{\frac{d\sqrt{1 + 4\nu^2}}{-2\kappa}},\\d = \frac{\sqrt{1 + 4\nu^2} - 1}{2\nu},\\\omega = -\frac{d\left(1 + 4\nu^2\right)}{2\nu}.\tag{43}$$

The convergence rates and the numerical L<sup>2</sup> error are listed in Figure 1 for several different values of ν, confirming optimal O hNþ<sup>1</sup> � � order of convergence across.

Figure 1. The rate of convergence for the solving the nonlinear Ginzburg-Landau equation in Example 4.2.

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

to advance from u<sup>n</sup>

∂u ∂t

Δt ≤ CΔx<sup>α</sup>

with

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

124 Differential Equations - Theory and Current Research

min ð Þ 0 < C < 1 is used to ensure stability.

<sup>η</sup> <sup>¼</sup> <sup>1</sup> 2

The exact solution u xð Þ¼ ; <sup>t</sup> a xð Þeidln ð Þ� a xð Þ <sup>i</sup>ω<sup>t</sup> where

a xð Þ¼ Fsech xð Þ, F ¼

Example 4.1 We consider the following linear Ginzburg-Landau equation

, <sup>κ</sup> ¼ � <sup>ν</sup> <sup>3</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> <sup>4</sup>ν<sup>2</sup> <sup>p</sup> �2κ

Figure 1. The rate of convergence for the solving the nonlinear Ginzburg-Landau equation in Example 4.2.

s

values of ν, confirming optimal O hNþ<sup>1</sup> � � order of convergence across.

<sup>1</sup> <sup>þ</sup> <sup>4</sup>ν<sup>2</sup> <sup>p</sup> � <sup>1</sup> � �

, d ¼

The convergence rates and the numerical L<sup>2</sup> error are listed in Figure 1 for several different

<sup>h</sup> , separated by the time step, Δt. In our examples, the condition

2 2 <sup>þ</sup> <sup>9</sup>ν<sup>2</sup> ð Þ , <sup>ζ</sup> ¼ �1, <sup>γ</sup> <sup>¼</sup> <sup>0</sup>: (42)

<sup>2</sup><sup>ν</sup> , <sup>ω</sup> ¼ � <sup>d</sup> <sup>1</sup> <sup>þ</sup> <sup>4</sup>ν<sup>2</sup> � �

<sup>2</sup><sup>ν</sup> : (43)

� ð Þ ν þ iη Δu þ ð Þ κ þ iζ u ¼ 0, x∈½ � �20; 20 , t∈ ð � 0; 0:5 , uxð Þ¼ ; 0 u0ð Þx , (41)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> <sup>4</sup>ν<sup>2</sup> <sup>p</sup> � <sup>1</sup>

$$
\mu(\mathbf{x}, \mathbf{0}) = \mathbf{e}^{-\mathbf{x}^2},
\tag{44}
$$

with parameters ν ¼ 1, κ ¼ 1, η ¼ 1, ζ ¼ 2, x ∈½ � �10; 10 . We consider cases with N = 2 and K = 40 and solve the equation for several different values of γ. The numerical solution uhð Þ x; t for γ ¼ 2, 1, 0, � 1, � 2 is shown in Figures 2 and 3. The parameter γ will affect the wave shape. From these figures, it is obvious that the solution decays rapidly with time evolution especially for γ < 0 and the parameter γ dramatically affects the wave shape.

Figure 2. Numerical results for the nonlinear Ginzburg-Landau equation in Example 4.2.

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Figure 3. Numerical results for the nonlinear Ginzburg-Landau equation with γ = 2 in Example 4.2.
