High-order Accurate Numerical Methods For Nonlinear Dirac Equations

The nonlinear Dirac(NLD)equation can be used as a nonlinear model to describe the spin-1/2 particles.It has solitary wave solution or particle-like solution,which contains rich nonlinear phenomena.Under certain conditions,the NLD equation satisfies the conservations of the total charge,the linear momentum and the total energy.The purpose of this thesis is to study high-order accurate numerical methods for the NLD equation with a general self-interaction.The numerical methods include the operator-compensation(OC)methods,the compact difference(CD)methods and the discontinuous Galerkin(DG)methods.The OC method is derived by extending the second-order central difference quo-tient operator.By adding grid nodes,the spatial accuracy of the numerical scheme can reach any even order.In this thesis,the fully discrete OC schemes based on the second-order accurate Crank-Nicolson method and the Strang time-splitting method for the one-dimensional(1D)and two-dimensional(2D)NLD equations are present-ed,denoted by the CNOC scheme and the TSOC scheme,respectively.We prove theoretically that the semi-discrete OC scheme and the CNOC scheme conserve both the discrete charge and the discrete energy,while the TSOC scheme only conserves the discrete charge.It is also proved that these schemes are unconditionally(linearly)stable.The accuracy and the conservative properties of the 1D and 2D schemes are verified by the numerical examples,the results are consistent with the theoretical analysis.In addition,the numerical examples are used to simulate the collision of the 1D solitary waves and the dynamics of the 2D solitary waves.In the collision of the 1D solitary waves,the CNOC scheme can also conserve the discrete charge and the discrete energy.The CD method is attractive because it can use fewer grid points to achieve high-order accuracy.This thesis presents the fourth-order CD scheme,the three-point sixth-order combined CD(CCD)scheme and their linearized schemes for the 1D and 2D NLD equations.It is theoretically proved that these schemes are unconditionally(linearly)stable.Numerical examples verify the accuracy of the schemes for both the 1D and 2D cases.From the numerical point of view,the conservative property of the numerical scheme for the 1D case is studied,from which it can be seen that although there is no strictly theoretical proof,the numerical results show that the four schemes conserve the discrete charge very well.Moreover,the CD scheme and the CCD scheme have good ability of keeping the discrete energy.Numerical examples are also used to simulate the collision of the 1D solitary waves and the dynamics of the 2D solitary waves.The DG method is a kind of the finite element method proposed in the 1970s.It uses the discontinuous piecewise polynomials to approximate the true solution,so it is suitable to deal with the problem in the complex domain.Moreover,it has the advantages of high accuracy and effectively capturing the discontinuities.In this thesis,the RKDG scheme for solving the 1D NLD equation[Shao&Tang,Discrete Cont.Dyn.Sys.B,6(2006)]is extended to the 2D case,and it is proved that the 2D semi-discrete DG scheme satisfies the entropy inequality.In addition,we also develop the DG schemes based on the fourth-order accurate Lax-Wendroff type time discretization and the two-stage fourth-order accurate time discretization,denoted by the LWDG scheme and TSDG scheme,respectively.By estimating the computational complexity of those three DG schemes in the 1D case,we find that the TSDG scheme needs the least CPU time,followed by the RKDG scheme,while the LWDG scheme needs the most CPU time.Numerical examples show that the DG schemes can achieve the expected accuracy for both the 1D and 2D case.The conservative properties,the collision of the 1D solitary waves and the dynamics of the 2D solitary waves are simulated.Numerical examples show that:(?)Compared with the finite difference schemes,the L2 and L? errors of the DG schemes increase slowly,and the errors of the RKDG scheme increase more slowly than the other two DG schemes,so the RKDG scheme is very suitable for the long-time simulation;(?)The inelastic collision in the binary and ternary collisions is observed again in the 1D case,meanwhile,the inelastic collision is also observed in the quaternary collision;(?)For the 2D case,the long-time oscillation state produced by two standing waves is observed;(iv)For the 2D standing wave solution with the frequency ?=0.12,after a certain time,the charge density changes periodically with "circular ring-elliptical ring-circular ring",but this phenomenon is not observed when ?=0.8;(v)For the 2D case,the charge density shows a breathing state when ?=0.94 and ?=2(? is the number of nonlinear terms),but it is not observed when ?=1.In addition,based on the existing standing wave solution,the traveling wave so-lution of the 2D NLD equation is obtained by the Lorentz transformation(the relative velocity of the two inertial reference systems is parallel to the x-axis).Different from the standing wave solution,the charge density of the traveling wave solution loses symmetry in the y-direction.The numerical examples simulate the time evolution of the charge density of the traveling wave solution.