Investigation of how Reflected Radiation due to Non-Uniform Grids Perturbs Solutions Computed with Finite Difference Methods

We investigate the behavior of solutions to the 1D NonLinear Coupled Mode Equations (NLCME) with a Finite Difference (FD) method. The FD approach is used to render the NLCMEs a system of Ordinary Differential Equations (ODE) that is then solved with a splitting method. The high frequency components of the solution moving from a fine grid into a coarse grid cannot be resolved on the coarse grid and are reflected back into the fine region. Collision of two gap solitons produces high frequency radiation that travels faster than the gap solitons. This report investigates to what extend this reflected radiation perturbs the solution in the fine gird region and how fine the grid in the coarse region should be to neglect the resulting error. We observe that the properties of the reflected radiation depend on the FD approximation. We also test the conservation energy and Hamiltonian and compare it to Runge-Kutta methods, as well as the order in the timestep dt and the grid spacing h of the used method. We observe superconvergence, i.e., in the L-norm the order was 4 when we expect it to be of second order. Additionally, with our used method we could not reproduce the results from Ref. [1], as for example the merger of two gap solitons. This is probably due to the dissipative nature of the upwinding FD approximation. keywords: FD, Hamiltonian, Non-Uniform Grid, NLCME, gap solitons, collision