Structure-preserving Algorithms For The Klein-Gordon-Dirac Equation With The Small Coupling Constant

In recent years,the stability and convergence of numerical methods for the KleinGordon-Dirac(KGD)equation have received increasing attention from scholars.But as far as we know,there has not been energy-preserving exponential wave integrator method and relevant long time analysis for the KGD equation.For the the weakly coupled KGD equation(the coupling constant ε∈(0,1])with periodic boundary conditions,this paper proposes two novel time symmetric and structure-preserving exponential wave integral Fourier pseudo-spectral(TSSPEWIFP Ⅰ and TSSPEWIFP Ⅱ)methods in Chapters 2 and 3.Both of these methods use the exponential wave integration method in time and the Fourier pseudo-spectral method in space.The numerical schemes we obtained not only preserve the time symmetry and the energy conservation in discrete level,but also preserve the modified discrete mass in the TSSPEWIFP Ⅱ method.Through rigorous theoretical analysis,it is proved that the uniform error bounds of the two numerical schemes we obtained at O(hm0+ε1-βτ2)up to the time at O(1/εβ)with β∈[0,1]where h and τ are the mesh size and time step,and m0 depends on the regularity conditions.Error analysis tools include the cut-off technique and the standard energy method,the correctness of the conclusion is verified by numerical experiments in Chapter 4.In Chapter 5,I also extend the TSSPEWIFP Ⅰ method and the TSSPEWIFP Ⅱ method to the two-dimensional KGD equation and the oscillatory KGD equation,the correctness of the conclusion is verified by numerical experiments.Fast Fourier transform algorithm is used in the calculation,which greatly improves the operation efficiency.