Multi-symplectic Runge-Kutta-Nyström methods for nonlinear Schrödinger equations with variable coefficients

In this paper, we investigate multi-symplectic Runge-Kutta-Nyström (RKN) methods for nonlinear Schrödinger equations. Concatenating symplecitc Nyström methods in spatial direction and symplecitc Runge-Kutta methods in temporal direction for nonlinear Schrödinger equations leads to multi-symplecitc integrators, i.e. to numerical methods which preserve the multi-symplectic conservation law (MSCL), we present the corresponding discrete version of MSCL. We show that the multi-symplectic RKN methods preserve not only the global symplectic structure in time, but also local and global discrete charge conservation laws under periodic boundary conditions. Lower and higher order multi-symplectic RKN methods are utilized in numerical experiments. The errors of numerical solutions, the numerical errors of discrete energy, discrete momentum and discrete charge are exhibited. The precise conservation of discrete charge under the multi-symplectic RKN discretizaitons is attested numerically. By comparing with non-multi-symplectic methods, some numerical superiorities of the multi-symplectic RKN methods are exhibited.