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An asymptotic preserving scheme for the Schrödinger equation in the semiclassical limit. (English) Zbl 1128.65064
Summary: This Note is devoted to the discretization of the fluid formulation of the Schrödinger equation (the Madelung system). We explore both the discretization of the system in Eulerian coordinates and Lagrangian coordinates. We propose schemes for these two formulations which are implicit in the mass flux term. This feature allows us to show that these schemes are asymptotic preserving, i.e., they provide discretizations of the semi-classical Hamilton-Jacobi equation when the scaled Planck constant $$\epsilon$$ tends to 0. An analysis performed on the linearized systems also shows that they are asymptotically stable i.e. their stability condition remains bounded as $$\epsilon$$ tends to 0. Numerical simulations are given; they confirm that the considered schemes allow us to numerically bridge the quantum and semi-classical scales.

##### MSC:
 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35Q55 NLS equations (nonlinear Schrödinger equations) 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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