Variable-step mass-conserving numerical scheme for coupled nonlinear Schrödinger-Helmholtz equations

Structure-preserving numerical schemes for the time integer order Schrödinger-Helmholtz equations have been extensively investigated. However, relevant research on the two-dimensional time-fractional coupled nonlinear Schrödinger-Helmholtz equations remain relatively scarce. In view of this, a linearized variable-step structure-preserving scheme is proposed for solving these equations. Specifically, the spatial derivatives are approximated by the finite element method on unstructured meshes, while the temporal fractional derivatives are discretized via the fractional BDF2 scheme on nonuniform grids. A local energy conservation law and a local mass conservation law for the time-fractional coupled nonlinear Schrödinger-Helmholtz equations are proven by adopting the fractional Leibniz formulas on the complex field. Furthermore, the proposed scheme is proven to satisfy the discrete local mass conservation law without any restrictions. To improve computational efficiency, a fast algorithm is constructed. Finally, several numerical examples are presented to verify the accuracy of the structure-conserved schemes and the high efficiency of the fast algorithm.