Hermite Spectral Methods And Its Applications In Ginzburg-landau-schrodinger Equation

The numerical methods to solve partial di?erential equations mainly include ?nite difference methods, the ?nite element methods, and spectral methods. Of them, spectral methods are widely used for their high accuracy and high stability, and play a more and more important role in computational ?uid mechanics, superconductivity researches, numerical simulation of non-linear optics, and other ?elds. In This paper the numerical methods of partial di?erential equations in all-unbounded domains is studied. Finite di?erence methods and the ?nite element method would produce arti?cial boundary errors from reducing unbounded domains into bounded domains and there are some defects in rational spectral methods and projection methods, so it is well-reasoned to choose the generalized Hermite spectral method de?ned in all-unbounded domains in this paper.This paper introduced the orthogonal Hermite polynomial and generalized Hermite function with a compression factor, and then studied the nature of the generalized Hermite function and the related problems. It used the generalized Hermite spectral method to simulate the numerical solutions to second-order elliptic equations respectively under three forms, i.e. exponential decay, algebraic decay, and oscillatory algebraic decay. The numerical experiment shows that all the solutions under the three forms possess spectral accuracy, and the exponential decay has the best approximating e?ect; and that choosing an appropriate compression factor can improve the accuracy of numerical solutions.Time-splitting method is a simple, e?ective, and easy-to-realize numerical method. This paper introduced two time-splitting discrete forms of Ginzburg-Landau-Schr¨odinger equation(GLSE): time-splitting-di?erence-generalized Hermite spectral method and time-splitting term-matching generalized Hermite spectral method. The term-matching generalized Hermite spectral method makes full use of generalized Hermite function’s own characteristics and both the linear part and nonlinear part, divided by using this method, can be solved accurately and respectively; therefore, it is a simple and e?ective new numerical method with high accuracy and good stability. In addition, proved the stability and error estimation of GLSE’s semi-discrete term-matching generalized Hermite spectral method. At last, this paper carried out a numerical experiment to the two forms of GLSE, and the result proves that the accuracy of time-splitting term-matching generalized Hermite spectral method is obviously higher than that of time-splitting di?erence generalized Hermite spectral method and that choosing an appropriate compression factor can also improve the approximating accuracy of numerical solutions. Therefore, this method is a recommendable, simple and e?ective numerical method to solve di?erential equations, which possesses spectral accuracy as well.