Error Estimate of a New Conservative Finite Difference Scheme for the Klein-Gordon-Dirac System

Abstract

In this paper, we derive and analyze a conservative Crank-Nicolson-type finite difference scheme for the Klein-Gordon-Dirac (KGD) system. Differing from the derivation of the existing numerical methods given in literature where the numerical schemes are proposed by directly discretizing the KGD system, we translate the KGD equations into an equivalent system by introducing an auxiliary function, then derive a nonlinear Crank-Nicolson-type finite difference scheme for solving the equivalent system. The scheme perfectly inherits the mass and energy conservative properties possessed by the KGD, while the energy preserved by the existing conservative numerical schemes expressed by two-level's solution at each time step. By using energy method together with the 'cut-off' function technique, we establish the optimal error estimate of the numerical solution, and the convergence rate is O(τ 2 + h 2 ) in l ∞ -norm with time step τ and mesh size h. Numerical experiments are carried out to support our theoretical conclusions.

Improved error estimates of the time‐splitting methods for the long‐time dynamics of the Klein–Gordon–Dirac system with the small coupling constant

“…In practical application, efficient numerical methods are very necessary. Existing numerical methods include finite difference (FD) and compact FD (CFD) methods [8,12,31,41,44], exponential wave integrator (EWI) methods [27,44], time-splitting (TS) methods [42,45]. Generally speaking, compared with the FD and CFD methods, these EWI methods and TS methods perform better in numerical accuracy and regularity requirement.…”