A high-order nodal discontinuous Galerkin method for nonlinear fractional Schrödinger type equations
For researchers in numerical methods for fractional PDEs, this provides a stable, high-order scheme, but the approach is incremental (applying known DG techniques to a specific equation class).
The authors propose a nodal discontinuous Galerkin method for nonlinear Riesz space fractional Schrödinger equations, proving L² stability and optimal convergence order O(h^{N+1}), confirmed by numerical experiments.
We propose a nodal discontinuous Galerkin method for solving the nonlinear Riesz space fractional Schrödinger equation and the strongly coupled nonlinear Riesz space fractional Schrödinger equations. These problems have been expressed as a system of low order differential/integral equations. Moreover, we prove, for both problems, $L^{2}$ stability and optimal order of convergence $O(h^{N+1})$, where $h$ is space step size and $N$ is polynomial degree. Finally, the performed numerical experiments confirm the optimal order of convergence.