A spacetime DPG method for the Schrodinger equation
This work provides a new numerical method for solving the Schrödinger equation, particularly for capturing irregular solutions, which is relevant for computational physics and engineering.
The paper proposes a spacetime Discontinuous Petrov Galerkin (DPG) method for the linear time-dependent Schrödinger equation, proving well-posedness for two variational formulations and demonstrating convergence for the ultraweak formulation. Numerical experiments for pulse propagation in dispersive optical fibers are reported.
A spacetime Discontinuous Petrov Galerkin (DPG) method for the linear time-dependent Schrodinger equation is proposed. The spacetime approach is particularly attractive for capturing irregular solutions. Motivated by the fact that some irregular Schrodinger solutions cannot be solutions of certain first order reformulations, the proposed spacetime method uses the second order Schrodinger operator. Two variational formulations are proved to be well-posed: a strong formulation (with no relaxation of the original equation) and a weak formulation (also called the ultraweak formulation, that transfers all derivatives onto test functions). The convergence of the DPG method based on the ultraweak formulation is investigated using an interpolation operator. A standalone appendix analyzes the ultraweak formulation for general differential operators. Reports of numerical experiments motivated by pulse propagation in dispersive optical fibers are also included.