The Study Of The Second-order Numerical Scheme Of Ginzburg-Landau Equation And Its Point-wise Error Estimate

Fractional calculus theory is the generalization and extension of the integral calculus theory.As an important branch of mathematics,it has some special advantages and extensive application prospects in many scientific fields such as biological engineering,physics,chemistry,mechanics,signal processing and rheological properties of rocks.Fractional Ginzburg-Landau equation is an important fractional differential equation which has at-tracted extensive attention in mathematical analysis and physical application.However,the analytical solutions of fractional Ginzburg-Landau equations,especially,two-dimensional space fractional Ginzburg-Landau equation are difficult to obtain or unavailable.Their convergence rate rate is very slow,and the calculation of these functions is very time-consuming in practical applications.Therefore,it is very important practical significance and necessity to study two-dimensional space fractional Ginzburg-Landau equation.The main purpose of this paper is to propose a class of BDF2-ADI finite difference method and a class of a three-level linearized finite difference method for two-dimensional nonlinear space fractional complex Ginzburg-Landau equation.The convergence and stability of BDF2-ADI difference scheme under L2-norm and the convergence and unconditional stability of the three-level linearized difference scheme under L?-norm are studied and analyzed in detail.The specific research content and main conclusions are composed of the following four chapters:In the first chapter,we introduce the research background and significance of fractional differential equations,the research status at home and abroad,the research content and main methods of this paper.In addition,the preliminary knowledge in the research is also presented.In the second chapter,we mainly study the two-dimensional nonlinear space fractional Ginzburg-Landau equation based on the second-order backward differential formula(BDF2),and also establish a class of new numerical scheme.Riesz fractional centered finite difference method is employed for the spatial discretization,Linearized technique is employed by the extrapolation.In addition,we also construct the multistep ADI scheme for the efficient numerical implementation.Furthermore,The uniqueness,convergence and stability of the solution of the numerical scheme under L2-norm are discussed.Finally,numerical examples are carried out to verify our theoretical results.In the third chapter,a class of three-level linearized difference scheme is derived for two-dimensional nonlinear space fractional complex Ginzburg-Landau equation.The discrete fractional Sobolev embedding theorem under H?,?-norm is proved,the unique solvability of the scheme is analyzed,and unconditional stability of the scheme is obtained.The most important work is to obtain the optimal point-wise error estimate of the convergence of the numerical scheme.Numerical experiments are implemented to confirm the theoretical findings.The last chapter,we briefly summarize the main research work of this paper,and point out the innovation of this paper as well as the research direction and content of the next step.