Optimal point-wise error estimate of a compact difference scheme for the Klein-Gordon-Schrödinger equation

In this paper, we propose a compact finite difference scheme for computing the Klein-Gordon-Schrödinger equation (KGSE) with homogeneous Dirichlet boundary conditions. The proposed scheme not only conserves the total mass and energy in the discrete level but also is linearized in practical computation. Except for the standard energy method, a new technique is introduced to obtain the optimal convergent rate, without any restriction on the grid ratios, at the order of O(h4+τ2) in the l∞-norm with time step τ and mesh size h. Finally, numerical results are reported to test the theoretical results.