An Efficient Numerical Method For Solving Nonlinear Dirac Equations

In this thesis,we develop a time splitting spectral method to solve time-dependent nonlinear Dirac system.The model has only first derivatives in time and space and no electromagnetic potential.The process of solving nonlinear Dirac equations is as follows.Firstly,applying splitting technique in time,the time-dependent nonlinear Dirac system is reduced to two parts:linear model and nonlinear model.Secondly,by Fourier-Galerkin(FG)spectral method to discretize the spatial variables of linear Dirac equations.During the discretized system.matrix diagonalization technique is applied to accomplish the solving process in(1+1)-dimensional,(2+1)-dimensional,(3+1)-dimensional Dirac equations.This thesis not only introduces the physical model transformation,system quantiza-tion,forms of conserved quantities,proof of conservation laws,but also introduces the Sequential splitting method and the Strang splitting method respectively and analyzes their advantages and disadvantages.Compared with conventional differ-ence method,Strang time splitting Fourier-Galerkin spectral method(SFG)can achieve spectral accuracy in space,the second order precision in time and increase computational speed remarkably.Finally,ample numerical experiments confirm the spectral accuracy and efficiency of this method.