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 u x ‾ , u xx in the
NSANLSE. If one uses classical high order compact schemes to approximate
u xx and 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.