A compact finite difference scheme for the nonlinear Schrödinger equation with wave operator

In this paper, a compact finite difference scheme is presented for an periodic initial value problem of the nonlinear Schrödinger (NLS) equation with wave operator. This is a scheme of three levels with a discrete conservation law. The unconditional stability and convergence in maximum norm with order O(h4+τ2)O(h4+τ2) are proved by the energy method. A numerical experiment is presented to support our theoretical results.