Finite Difference Approximation For Quintic Ginzburg-Landau Equations

This paper discusses the numerical solutions of nonlinear Ginzburg-Laudau equations by finite difference methods. We construct difference schemes to different questions and prove the convergence of all difference schemes proposed in this article. Nonlinear Ginzburg-Laudau equation is one of the most important nonlinear evolution equation,which has a rich physical background and content. Meanwhile,it has been widely used in various different fields. Due to the exact solution of these equations are di?cult to be given, so study the numerical solutions of these equations are significant to understanding the practical meaning of these equations. This report includes four chapters.The first chapter is an introduction. The research background and current situation of the problem are briefly introduced, some denotations and lemmas and research results are described.In the second chapter, we research the Ginzburg-Laudau equations with quintic items ut= α0u + α1uxx+ α2|u|2u + α3|u|4u,(x, t) ∈ R ×(0, T ]u(x, 0) = u0(x), x ∈ R,u(x + 1, t) = u(x, t), x ∈ R, t ∈(0, T ],where αj= aj+ibj, α0= a0, aj, bjare real numbers, and a1= Re(α1) > 0 > Re(α3) = a3. Two linearized difference schemes are constructed. At the same time, we prove the convergence of these difference schemes by mathematical induction in L2 norm, through numerical results we can see that the order of convergence is O(τ2+ h2).In the third chapter, we continue to study with Ginzburg-Laudau which has initial value and periodic-boundary conditions. A compact difference scheme is proposed. As we all know, the most di?cult problem in the convergence analysis of difference equations is to obtain priori estimate in uniform norm. To solve this problem, this article transform point by point format to vector format, then through the knowledge of matrix to get the priori estimate in uniform norm of the numerical solution. We prove the convergence of the difference scheme in L∞ norm, and the order of convergence is O(τ2+ h4). At the last part of this chapter, some numerical results are shown to prove the difference method is e?cient.In the fourth chapter, we consider more general Ginzburg-Laudau as follows,ut= α0u + α1uxx+ α2|u|2u + α3|u|2ux+ α4u2ux+ α5|u|4u,(x, t) ∈ R ×(0, T ]u(x, 0) = u0(x), x ∈ R,u(x + 1, t) = u(x, t), x ∈ R, t ∈(0, T ],where αj= aj+ ibj, j = 1, 2, 5, αj= aj, j = 0, 3, 4, aj, bjare real numbers, and a1=Re(α1) > 0 > Re(α5) = a5. Contrast to the equation discussed in last chapter, this equation is more complicated since it has a first derivative about space. Through construct a proper di?erence scheme for the first derivative, and get uniform norm of the numerical solution, we prove the convergence of the di?erence scheme in L∞ norm, and the order of convergence is O(τ2+ h2).