Ginzburg-Landau-Schrodinger 方程的一致精确算法及其误差估计

Ginzburg-Landau-Schrodinger方程是一类典型的非线性耗散型发展方程，广泛应用于描述涡旋等多种物理现象。当前对该方程的算法研究主要针对非线性项不具有刚性的情形。然而在涡旋等应用中，方程中的刚性因子(小参数)接近零，此时非线性项将具有强刚性，方程的解在较小区段内发生剧烈变化，采用一般数值算法求解时需要使用比刚性因子更小尺度的步长，否则将产生较大误差甚至产生非物理解。本项目将研究时间步长不受刚性因子限制的一致精确的高效算法。我们首先从保持方程渐近性出发，分析几类经典算法的渐近保持性并构造可高效求解的高阶渐近保持格式。最后研究上述算法的一致误差估计并总结具有一致精度的算法特征，进一步设计新的一致精确算法并建立相关理论。本项目所获结果将丰富和发展耗散型发展方程的算法理论，并为实际应用提供算法支持和理论依据。

The Ginzburg-Landau-Schrodinger equation is a class of typical nonlinear dissipative evolution equations, which has wide applications in various physical fields including the vortices. So far, there are a lot of numerical algorithms for the equation with non-stiff nonlinearity. However, in the application of vortices, the nonlinear term becomes stiff when the stiffness parameter (small parameter) in the equation goes to zero. In this case, the solution of equation will have drastic changes in a small interval. So a naive numerical method requires the time step smaller than the scale of the stiffness parameter. The purpose of this project is to propose and analyze some uniformly accurate numerical methods so that the time step constraint is independent of the stiffness parameter. To this end, we firstly consider the asymptotic behavior of the equation, and study the ability in asymptotic-preserving of some classical methods，as well as construct some new asymptotic-preserving schemes. The ultimate aim is to analyze the uniform error estimates of above obtained methods and extract the features of algorithms with uniform accuracy, so as to design some new uniformly accurate numerical methods. The results obtained in the project will enhance the theory of numerical analysis of dissipative evolution equations, and provide the technical support and theoretical basis for applications.