Numerical approximation of quadratic observables of Schrödinger-type equations in the semi-classical limit

Markowich, Peter A.; Pietra, Paola; Pohl, Carsten

doi:10.1007/s002110050406

Cited by 104 publications

(106 citation statements)

References 14 publications

Supporting

Mentioning

103

Contrasting

Unclassified

Order By: Relevance

“…Moreover, these characteristic rays coincides with those obtained by the resolution of the Hamiltonian systems associated to the principal symbols (1.11) and (1.10). Our results extend the ones by Macià in [22] dealing with numerical approximations of the wave equation with variable coefficients on uniform meshes and the ones by Markowich-Pietra-Pohl in [26] in which several discrete schemes for the Schrödinger equation on uniform meshes are analyzed.…”

Section: The Wigner Transformsupporting

confidence: 80%

Propagation of 1D Waves in Regular Discrete Heterogeneous Media: A Wigner Measure Approach

Marica

Zuazua

2014

Found Comput Math

View full text Add to dashboard Cite

Abstract. In this article, we describe the propagation properties of the one-dimensional wave and transport equations with variable coefficients semi-discretized in space by finite difference schemes on non-uniform meshes obtained as diffeomorphic transformations of uniform ones. In particular, we introduce and give a rigorous meaning to notions like the principal symbol of the discrete wave operator and the corresponding bi-characteristic rays. The main mathematical tool we employ is the discrete Wigner transform, which, in the limit as the mesh size parameter tends to zero, yields the so-called Wigner (semi-classical) measure. This measure provides the dynamics of the bi-characteristic rays, i.e., the solutions of the Hamiltonian system describing the propagation, in both physical and Fourier spaces, of the energy of the solution to the wave equation. We show that, due to dispersion phenomena, the high-frequency numerical dynamics does not coincide with the continuous one. Our analysis holds for C 0,1 (R)-coefficients and non-uniform grids obtained by means of C 1,1 (R)-diffeomorphic transformations of a uniform one. We also present several numerical simulations that confirm the predicted paths of the space-time projections of the bi-characteristic rays. Based on the theoretical analysis and simulations, we describe some of the pathological phenomena that these rays might exhibit as, for example, their reflection before touching the boundary of the space domain. This leads, in particular, to the failure of the classical properties of boundary observability of continuous waves, arising in control and inverse problems theory.

show abstract

Section: The Wigner Transformsupporting

confidence: 80%

Propagation of 1D Waves in Regular Discrete Heterogeneous Media: A Wigner Measure Approach

Marica

Zuazua

2014

Found Comput Math

View full text Add to dashboard Cite

show abstract

“…Generally 1) they require severe stability constraints on the mesh sizes, 2) they do not conserve the total particle number, 3) they are not time transverse invariant. For a mathematical analysis of FD-methods for Schrödinger type equations in semiclassical regimes we refer to [43,44].…”

Section: Numerical Approximationmentioning

confidence: 99%

Numerical solution of the Gross–Pitaevskii equation for Bose–Einstein condensation

Bao

Jaksch²,

Markowich³

2003

Journal of Computational Physics

Self Cite

526

582

View full text Add to dashboard Cite

We study the numerical solution of the time-dependent Gross-Pitaevskii equation (GPE) describing a Bose-Einstein condensate (BEC) at zero or very low temperature. In preparation for the numerics we scale the 3d GrossPitaevskii equation and obtain a four-parameter model. Identifying 'extreme parameter regimes', the model is accessible to analytical perturbation theory, which justifies formal procedures well known in the physical literature: reduction to 2d and 1d GPEs, approximation of ground state solutions of the GPE and geometrical optics approximations. Then we use a time-splitting spectral method to discretize the time-dependent GPE. Again, perturbation theory is used to understand the discretization scheme and to choose the spatial/temporal grid in dependence of the perturbation parameter. Extensive numerical examples in 1d, 2d and 3d for weak/strong interactions, defocusing/focusing nonlinearity, and zero/nonzero initial phase data are presented to demonstrate the power of the numerical method and to discuss the physics of Bose-Einstein condensation. *

show abstract

“…If a constant α is added to the potential V , then the discrete wave functions U ε,n+1 j obtained from TS-Cosine4 or TS-Cosine2 get multiplied by the phase factor e −iα(n+1)k/ε , which leaves the discrete quadratic observables unchanged. This property does not hold for finite difference schemes [25,26].…”

mentioning

confidence: 97%

“…In [25,26], Markowich et al studied the finite difference approximation of the linear Schrödinger equation with small ε and zero far-field condition. Their results show that, for the best combination of the time and space discretizations, one needs the following constraints in order to guarantee good approximations to all (smooth) observables for ε small [25,26]: mesh size h = o(ε) and time step k = o(ε). The same or more severe meshing constraint is required by the finite difference approximation of the NLS in the semi-classical regime with zero far-field condition, i.e.…”

mentioning

confidence: 99%

Numerical Methods for the Nonlinear Schrödinger Equation with Nonzero Far-field Conditions

Bao¹

2004

Methods and Applications of Analysis

View full text Add to dashboard Cite

show abstract

Numerical approximation of quadratic observables of Schrödinger-type equations in the semi-classical limit

Cited by 104 publications

References 14 publications

Propagation of 1D Waves in Regular Discrete Heterogeneous Media: A Wigner Measure Approach

Propagation of 1D Waves in Regular Discrete Heterogeneous Media: A Wigner Measure Approach

Numerical solution of the Gross–Pitaevskii equation for Bose–Einstein condensation

Numerical Methods for the Nonlinear Schrödinger Equation with Nonzero Far-field Conditions

Contact Info

Product

Resources

About