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.
- Ginzburg-Landau equation
- discontinuous Galerkin method
- error estimates
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 . The Taylor-Couette flow, Bénard convection  and plane Poiseuille flow  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 are real constants, . 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, 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 L2 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 and set
then, the Ginzburg-Landau problem can be rewritten as
We consider problem posed on the physical domain with boundary and assume that a nonoverlapping element such that
Now we introduce the broken Sobolev space for any real number r
We define the local inner product and norm
as well as the global broken inner product and norm
We define the jumps along a normal, , as
The numerical traces (u,s) are defined on interelement faces as the central fluxes
Let us discretize the computational domain into nonoverlapping elements, , and . We assume be the approximation of respectively, where the approximation space is defined as
where denotes the set of polynomials of degree up to N defined on the element . We define local discontinuous Galerkin scheme as follows: find , such that for all test functions ,
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 is simply a scalar and takes the value of +1 and −1 at the right and the left interface, respectively.
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. (L2 stability). The solution given by the LDG method defined by (13) satisfies
for any .
Proof. Set in (13) and consider the integration by parts formula , we get
Taking the real part of the resulting equation, we obtain
Removing the positive term , we obtain
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
For the error estimate, we define special projections and into . For all the elements, , are defined to satisfy
For the abovementioned special projections, we have, by the standard approximation theory , 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.
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
Summing over k, simplify by integration by parts and (9), we get
we can rewrite (29) as
provided c2 is sufficiently small such that , we obtain 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 , 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 . Numerical experiments demonstrate its numerical stability
where is the vector of unknowns, we can use the standard fourth-order four-stage explicit RK method (ERK)
to advance from to , separated by the time step, . In our examples, the condition is used to ensure stability.
Example 4.1 We consider the following linear Ginzburg-Landau equation
The exact solution where
The convergence rates and the numerical L2 error are listed in Figure 1 for several different values of , confirming optimal order of convergence across.
Example 4.2 We consider the nonlinear Ginzburg-Landau Eq. (1) with initial condition,
with parameters , . We consider cases with N = 2 and K = 40 and solve the equation for several different values of . The numerical solution for is shown in Figures 2 and 3. The parameter will affect the wave shape. From these figures, it is obvious that the solution decays rapidly with time evolution especially for and the parameter dramatically affects the wave shape.
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.