Uniform point-wise error estimates of semi-implicit compact finite difference methods for the nonlinear Schrodinger equation perturbed by wave operator

摘要

This study is devoted to analysis of semi-implicit compact finite difference (SICFD) methods for the nonlinear Schrodinger equation (NLS) perturbed by the wave operator (NLSW) with a perturbation strength described by a dimensionless parameter epsilon is an element of (0, 1]. Uniform l(infinity)-norm error bounds of the proposed SICFD schemes are built to give immediate insight on point-wise error occurring as time increases, and the explicit dependence of the mesh size and time step on the parameter a is also figured out. In the small is an element of regime, highly oscillations arise in time with O(epsilon(2))-wavelength. This highly oscillatory nature in time as well as the difficulty raised by the compact FD discretization make establishing the l(infinity)-norm error bounds uniformly in a of the SICFD methods for NLSW to be a very interesting and challenging issue. The uniform l(infinity)-norm error bounds in a are proved to be of O(h(4) + tau) and O(h(4) + tau(2/3)) with time step tau and mesh size h for well-prepared and ill-prepared initial data. Finally, numerical results are reported to verify the error estimates and show the sharpness of the convergence rates in the respectively parameter regimes. (C) 2014 Elsevier Inc. All rights reserved.