Combined high order compact schemes for non-self-adjoint nonlinear Schrödinger equations

Abstract

Some combined high order compact (CHOC) schemes are proposed for non-self-adjoint and nonlinear Schrödinger equation (NSANLSE). There are first order and second order spatial derivatives $\overline{u_x}$, $u_{xx}$ in the NSANLSE. If one uses classical high order compact schemes to approximate $u_{xx}$ and $\overline{u_x}$ separately, it will widen the bandwidth in practical coding due to matrix multiplication. This will partly counteract the advantages of high order compact. To overcome the deficiency, one solves the spatial derivatives simultaneously by combining them. In other words, it solves ${u_x}_j^n$ and ${u_{xx}}_j^n$ simultaneously in terms of $u_j$. The idea is applied to discretize NSANLSE in space. Two efficient numerical schemes are proposed for NSANLSE. The stability and convergence of the new schemes are analyzed theoretically. Numerical experiments are reported to verify the new schemes.