Spherical Harmonic-Generalized Laguerre Mixed Spectral Methods In Unbounded Domains

Spectral method is one of important numerical method for solving differential equations. The basic idea of Fourier spectral methods stems from 19th. But only in the past three decades,vareous spectral methods formed a branch of computational mathematics with strict theortical ananlysis. The fascinating merit of spectral method is its high accuracy. Because of this, spectral method has been applied successfully to computation of many problems arising in science, technology and economy, such as numerical simulations of many problems in heat conduction, fluid dynamics, quantum mechanics and financial mathematics and so on.In this paper, we investigate mainly a new generalized Laguerre spectral method and spherical harmonic-generalized Laguerre mixed spectral method for three dimensional unbounded domains and their applications.First, we propose a new generalized Laguerre polynomial orthogonal approximation which can be used for more practical problems. Moreover, the numerical solutions fit the exact solutions better so that it raises the numerical accuracy. We also develope the corresponding spherical harmonic-generalized Laguerre polynomial mixed spectral method and apply them successfully to computation of three dimensional problems, exterior problems and nonlinear problems.Next, we focus on new generalized Laguerre function orthogoal approximation. In this case, the numerical solution possess the same weight function as those of ori-gianl problem. Therefore they simulate the global properties of solutions of original problems very well. This technique also simplifies actual calcution and numerical analysis.We carry out many calculation including parallel computation. The numerical results demonstrate the spectral accuracy of proposed methods and the stablity of longtime calculation.