Fabricio Macià

On the consistency of MPS

Abstract The consistency of the Moving Particle Semi-implicit (MPS) method in reproducing the gradient, divergence and Laplacian differential operators is discussed in the present paper. Its relation to the Smoothed Particle Hydrodynamics (SPH) method is rigorously established. The application of the MPS method to solve the Navier–Stokes equations using a fractional step approach is treated, unveiling inconsistency problems when solving the Poisson equation for the pressure. A new corrected MPS method incorporating boundary terms is proposed. Applications to one dimensional boundary value Dirichlet and mixed Neumann–Dirichlet problems and to two-dimensional free-surface flows are presented.

A Boundary Integral SPH Formulation Consistency and Applications to ISPH and WCSPH

One of the historical problems appearing in SPH formulations is the inconsistencies coming from the inappropriate implementation of boundary conditions. In this work, this problem has been investigated; instead of using typical methodologies such as extended domains with ghost or dummy particles where severe inconsistencies are found, we included the boundary terms that naturally appear in the formulation. First, we proved that in the 1D smoothed continuum formulation, the inclusion of boundary integrals allows for a consistent O (h) formulation close to the boundaries. Second, we showed that the corresponding discrete version converges to a certain solution when the discretization SPH parameters tend to zero. Typical tests with the first and second derivative operators confirm that this boundary condition implementation works consistently. The 2D Poisson problem, typically used in ISPH, was also studied, obtaining consistent results. For the sake of completeness, two practical applications, namely, the duct flow and a sloshing tank, were studied with the results showing a rather good agreement with former experiments and previous results. Subject Index: 024

High-frequency propagation for the Schrödinger equation on the torus

The main objective of this paper is understanding the propagation laws obeyed by high-frequency limits of Wigner distributions associated to solutions to the Schrodinger equation on the standard d-dimensional torus Td. From the point of view of semiclassical analysis, our setting corresponds to performing the semiclassical limit at times of order 1/h, as the characteristic wave-length h of the initial data tends to zero. It turns out that, in spite that for fixed h every Wigner distribution satisfies a Liouville equation, their limits are no longer uniquely determined by those of the Wigner distributions of the initial data. We characterize them in terms of a new object, the resonant Wigner distribution, which describes high-frequency effects associated to the fraction of the energy of the sequence of initial data that concentrates around the set of resonant frequencies in phase-space T*Td. This construction is related to that of the so-called two-microlocal semiclassical measures. We prove that any limit μ of the Wigner distributions corresponding to solutions to the Schrodinger equation on the torus is completely determined by the limits of both the Wigner distribution and the resonant Wigner distribution of the initial data; moreover, μ follows a propagation law described by a family of density-matrix Schrodinger equations on the periodic geodesics of Td. Finally, we present some connections with the study of the dispersive behavior of the Schrodinger flow (in particular, with Strichartz estimates). Among these, we show that the limits of sequences of position densities of solutions to the Schrodinger equation on T2 are absolutely continuous with respect to the Lebesgue measure.

Wigner Measures in the Discrete Setting: High-Frequency Analysis of Sampling and Reconstruction Operators

The goal of this article is to determine how the oscillation and concentration effects developed by a sequence of functions in

Semiclassical measures for the Schrödinger equation on the torus

\mathbb{R}^{d}

Wigner measures and observability for the Schrödinger equation on the disk

are modified by the action of sampling and reconstruction operators on regular grids. Our analysis is performed in terms of Wigner and defect measures, which provide a quantitative description of the high-frequency behavior of bounded sequences in

Semiclassical measures and the Schrödinger flow on Riemannian manifolds

L^{2}( \mathbb{R}^{d})

Two-species-coagulation approach to consensus by group level interactions

. We actually present explicit formulas that make possible the computation of such measures for sampled/reconstructed sequences. As a consequence, we are able to characterize sampling and reconstruction operators that preserve or filter the high-frequency behavior of specific classes of sequences. The proofs of our results rely on the construction and manipulation of Wigner measures associated to sequences of discrete functions.

The Effect of Group Velocity in the Numerical Analysis of Control Problems for the Wave Equation

In this article, the structure of semiclassical measures for solutions to the linear Schrödinger equation on the torus is analysed. We show that the disintegration of such a measure on every invariant lagrangian torus is absolutely continuous with respect to the Lebesgue measure. We obtain an expression of the Radon-Nikodym derivative in terms of the sequence of initial data and show that it satisfies an explicit propagation law. As a consequence, we also prove an observability inequality, saying that the L-norm of a solution on any open subset of the torus controls the full L-norm.

Some Remarks on Quantum Limits on Zoll Manifolds

We analyse the structure of semiclassical and microlocal Wigner measures for solutions to the linear Schrödinger equation on the disk, with Dirichlet boundary conditions. Our approach links the propagation of singularities beyond geometric optics with the completely integrable nature of the billiard in the disk. We prove a “structure theorem”, expressing the restriction of the Wigner measures on each invariant torus in terms of second-microlocal measures. They are obtained by performing a finer localization in phase space around each of these tori, at the limit of the uncertainty principle, and are shown to propagate according to Heisenberg equations on the circle. Our construction yields as corollaries (a) that the disintegration of the Wigner measures is absolutely continuous in the angular variable, which is an expression of the dispersive properties of the equation; (b) an observability inequality, saying that the