Unconditional $$L^{\infty }$$ L ∞ -convergence of two compact conservative finite difference schemes for the nonlinear Schrödinger equation in multi-dimensions

Abstract

This paper is concerned with the unconditional and optimal $$L^{\infty }$$ -error estimates of two fourth-order (in space) compact conservative finite difference time domain schemes for solving the nonlinear Schrodinger equation in two or three space dimensions. The fact of high space dimension and the approximation via compact finite difference discretization bring difficulties in the convergence analysis. The two proposed schemes preserve the total mass and energy in the discrete sense. To establish the optimal convergence results without any constraint on the time step, besides the standard energy method, the cut-off function technique as well as a ‘lifting’ technique are introduced. On the contrast, previous works in the literature often require certain restriction on the time step. The convergence rate of the proposed schemes are proved to be of $$O(h^4+\tau ^2)$$ with time step $$\tau $$ and mesh size h in the discrete $$L^{\infty }$$ -norm. The analysis method can be directly extended to other finite difference schemes for solving the nonlinear Schrodinger-type equations. Numerical results are reported to support our theoretical analysis, and investigate the effect of the nonlinear term and initial data on the blow-up solution.