An asymptotic preserving approach for nonlinear Schrodinger equation in the semiclassical limit
This work provides a practical numerical scheme for quantum-classical transition problems, particularly relevant for computational physics and semiclassical analysis.
The authors developed an asymptotic preserving numerical method for the nonlinear Schrödinger equation in the semiclassical limit, enabling accurate recovery of position and current densities with mesh and time steps independent of the Planck constant, achieving convergence rates consistent with theory before shock formation in the limiting Euler equation.
We study numerically the semiclassical limit for the nonlinear Schroedinger equation thanks to a modification of the Madelung transform due to E.Grenier. This approach is naturally asymptotic preserving, and allows for the presence of vacuum. Even if the mesh size and the time step do not depend on the Planck constant, we recover the position and current densities in the semiclassical limit, with a numerical rate of convergence in accordance with the theoretical results, before shocks appear in the limiting Euler equation. By using simple projections, the mass and the momentum of the solution are well preserved by the numerical scheme, while the variation of the energy is not negligible numerically. Experiments suggest that beyond the critical time for the Euler equation, Grenier's approach yields smooth but highly oscillatory terms.