A mass and energy conservative fourth-order compact difference scheme for the Klein-Gordon-Dirac equations\(^{\star }\)

This paper is concerned with numerical solution of the two-dimensional Klein-Gordon-Dirac equations by a fourth-order compact finite difference method in space and an energy-preserving Crank-Nicolson-type discretization in time. For convenience of illustrating the conservative properties and investigating the convergence results, we convert the component-wise form of the proposed scheme into an equivalent matrix-vector form. By using the mathematical induction argument and standard energy method, we establish the optimal error estimates under condition \(\tau \le \frac{1}{|\mathrm {ln}(h)|}\) with time step \(\tau\) and mesh size h. Compared with the condition \(\tau =O(h^{\frac{1}{2}})\) required by existing results in literature for two-dimensional case, this greatly relaxes the dependence of the time step on the grid size. The convergence order of the scalar \(\phi\) and the 2-spinor \(\psi\) is of \(O(\tau ^{2}+h^{4})\) in the maximum norm and the discrete \(H^{1}\)-norm, respectively. In addition, by using the orthogonal diagonalization technique, a fast solver is designed to solve the proposed scheme. Several numerical results are reported to verify the error estimates and the discrete conservation laws.