Error Estimates of Finite Difference Methods for the Biharmonic Nonlinear Schrodinger Equation

We present two finite difference time domain methods for the biharmonic nonlinear Schrodinger equation (BNLS) by reformulating it into a system of second-order partial differential equations instead of a direct discretization, including a second-order conservative Crank-Nicolson finite difference (CNFD) method and a second-order semi-implicit finite difference (SIFD) method. The CNFD method conserves the mass and energy in the discretized level, and the SIFD method only needs to solve a linear system at each time step, which is more efficient. By energy method, we establish optimal error bounds at the order of O (h(2) + tau(2)) in both L-2 and H-2 norms for both CNFD and SIFD methods, with mesh size h and time step tau. The proof of the error bounds are mainly based on the discrete Gronwall's inequality and mathematical induction. Finally, numerical results are reported to confirm our error bounds and to demonstrate the properties of our schemes.