A new numerical scheme for the nonlinear Schrödinger equation with wave operator

In this paper, a new compact finite difference scheme is proposed for a periodic initial value problem of the nonlinear Schrödinger equation with wave operator. This is an explicit scheme of four levels with a discrete conservation law. The unconditional stability and convergence in maximum norm with order $$O(h^{4}+\tau ^{2})$$O(h4+τ2) are verified by the energy method. Those theoretical results are proved by a numerical experiment and it is also verified that this scheme is better than the previous scheme via comparison.

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