The Convergence Of A Difference Scheme For Nonlinear Ginzburg-Laudau Equations

This paper discusses the numerical solutions of nonlinear Ginzburg-Laudau equations by finite difference method.We construct difference schemes and prove the convergence of the difference schemes. Nonlinear Ginzburg-Laudau equations are one of the most active research topics,which are widely applied in various different fields.There are many methods to research the numerical solutions of the Ginzburg-Laudau equations, whereby finite difference method is widely used by the experts at home and abroad. This report consists 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 involving a quintic term with initial value and periodic-boundary conditions where αj=αj+ibj,α0=α0,and αj,bj are two real constants,This paper constructs difference schemes for the Ginzburg-Laudau equations and proves the convergence of the difference scheme in L∞norm and the order of convergence is O(τ2+h2), when α1=Re(α1)>0> Re(α3)=a3.In the third chapter, we research nonlinear Ginzburg-Laudau equations of one di-mention with initial value and periodic-boundary conditions where i2=-1,c1andc2are two real constants, w(x,t) is an unknown complex function and w0(x) is a given one. In this chapter, we construct a nonlinear compact difference scheme, we prove the existence of the solutions by Brouwer-type theorem and prove the convergence of this scheme by vector in L∞norm, whereby the order of convergence is O(τ2+h4). In the forth chapter, we research the high accuracy linearized difference scheme of the Ginzburg-Laudau equations that are researched in the third chapter. We prove the convergence of the difference scheme by mathematical induction in L∞norm,and the order of convergence is O(τ2+h4).