Abstract
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.
Keywords
- Ginzburg-Landau equation
- discontinuous Galerkin method
- stability
- error estimates
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
and periodic boundary conditions and
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
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
then, the Ginzburg-Landau problem can be rewritten as
We consider problem posed on the physical domain
Now we introduce the broken Sobolev space for any real number
We define the local inner product and
as well as the global broken inner product and norm
We define the jumps along a normal,
The numerical traces (
Let us discretize the computational domain
where
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
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.
Taking the real part of the resulting equation, we obtain
Removing the positive term
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
Subtracting (19) from the linear Ginzburg-Landau Eq. (13), we have the following error equation
For the error estimate, we define special projections
Denoting
For the abovementioned special projections, we have, by the standard approximation theory [21], that
where here and below
where the constant
Taking the real part of the resulting equation, we obtain
We take the test functions
we obtain
Summing over
we can rewrite (29) as
where
Using the definitions of the projections
From the approximation results (23) and Young’s inequality in (32), we obtain
and
Combining (34), (35), (36) and (30), we obtain
provided
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 [22], when the polynomials are of degree
where
to advance from
with
The exact solution
The convergence rates and the numerical
with parameters
5. Conclusions
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
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