A high-order nodal discontinuous Galerkin method for nonlinear fractional Schr'{o}dinger type equations

摘要

We propose a nodal discontinuous Galerkin method for solving the nonlinearRiesz space fractional Schr\"{o}dinger equation and the strongly couplednonlinear Riesz space fractional Schr\"{o}dinger equations. These problems havebeen expressed as a system of low order differential/integral equations.Moreover, we prove, for both problems, $L^{2}$ stability and optimal order ofconvergence $O(h^{N+1})$, where $h$ is space step size and $N$ is polynomialdegree. Finally, the performed numerical experiments confirm the optimal orderof convergence.