Highlights
Derivation of Langevin dynamics in a nonzero background flow field
Molecular dynamics is becoming more important to study the interaction of micro and macro scales. In fluid dynamics, the multi-scale approach often leads to analyzing or simulating the molecular system in a background flow field at constant temperature. Simulation based on stochastic Langevin molecular dynamics is an alternative for ergodic sampling at constant temperature. Langevin dynamics has been derived from different micro models: for instance in [G.W. Ford, M. Kac and P. Mazur, J. Math. Phys. 6 (1965), 504-515] and [R. Zwanzig, J. Stat. Phys. 9 (1973) 215-220] from a Hamiltonian system with a heavy particle coupled through a harmonic interaction potential to a heat bath particle system, where the stochasticity enters through Gibbs distributed initial heat bath configurations, and in [D. Dürr, S. Goldstein and J.L. Lebowitz Commun. Math. Phys. 78 (1981) 507-530] from a heavy particle colliding with an ideal gas heat bath, whose the initial particle configuration is described by a Poisson field. This highlight paper makes an important contribution to derivations of Langevin dynamics by extending the ideal gas model to non vanishing background flow.
Derivation of Langevin dynamics in a nonzero background flow field
Matthew Dobson, Frédéric Legoll, Tony Lelièvre and Gabriel Stoltz
ESAIM: M2AN 47 (2013) 1583-1626
DOI: http://dx.doi.org/10.1051/m2an/2013077
Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models
The numerical analysis of models and techniques of electronic structure computations is a long-term endeavor of prominent practical relevance given the ubiquity of such computations in several contemporary engineering and life sciences.
This highlight article in particular provides the first mathematical analysis for some discretization of the celebrated and commonly used Kohn-Sham model, within the local density approximation.
Claude Le Bris,
co-Editor-in-Chief
Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models
Eric Cancès, Rachida Chakir and Yvon Maday
ESAIM: M2AN 46 (2012) 341-388
DOI: http://dx.doi.org/10.1051/m2an/2011038
An error analysis of the multi-configuration time-dependent Hartree method of quantum dynamics
The multiconfiguration time-dependent Hartree (MCTDH) method is a commonly employed to approximate the N-body time-dependent Schrödinger equation. The resulting MCTDHF equations read as a coupled system of ordinary differential equations and nonlinear partial differential equations. Since its introduction by Meyer and coworkers in 1990, the MCTDH method has been successfully applied to the study of various quantum phenomena (photodissociation, rovibrational spectra of molecules, nonadiabatic transitions, tunneling, laser control of molecular processes, ...). This highlight article provides the first error analysis of the MCTDH approximation.
Eric Cancès
An error analysis of the multi-configuration time-dependent Hartree method of quantum dynamics
Dajana Conte and Christian Lubich
ESAIM: M2AN 44 (2010) 759-780
DOI: http://dx.doi.org/10.1051/m2an/2010018
Numerical simulation of blood flows through a porous interface
Mathematical models and computational methods are gaining much attention in medicine and biology. This highlight paper represents an excellent example in which advanced mathematical methods are applied to the study of a problem of medical and engineering relevance; i.e., the treatment of cerebral aneurysms. In particular, the authors apply mathematical homogenization to develop a model of blood flow through a sieve of metallic wires, which is applied in practice to prevent the rupture of aneurysms. This yields in the governing equations of blood flow a new term accounting for the sieve. An ad-hoc stabilized finite element discretization is then proposed and analyzed. Numerical experiments show that the computational method correctly captures the behavior of the sieve. This approach is used for the comparison of different treatment configurations in the frame of realistic models of cerebral aneurysms.
Alfio Quarteroni,
Associate Editor
Numerical simulation of blood flows through a porous interface
Miguel A. Fernandez, Jean-Frédéric Gerbeau and Vincent Martin
ESAIM: M2AN 42 (2008) 961-990
DOI: http://dx.doi.org/10.1051/m2an:2008031
A Roe-type scheme for two-phase shallow granular flows over variable topography
The paper treats a rather complex geophysical flow which consists of two-phase flow model for gravity-driven mixtures of solid material and fluid. It improves the recent Pitman-Le two-fluid model for debris flow by recovering the conservation of the mixture. The authors introduce a new model expressed in terms of a 4 × 4 hyperbolic system of equations, derive estimates of the four eigenvalues of the new system and analyze the possible breakdown of hyperbolicity. They proceed with the construction, analysis and implementation of a new numerical method for the proposed system. To overcome the difficulties of non-conservative products and stiff forcing terms, they generalize a Roe-type scheme to deal with nonlinear wave propagation. The resulting scheme is shown to be well-balanced for equilibrium as well as being very robust for moving equilibrium.
Eitan Tadmor,
Associate Editor
A Roe-type scheme for two-phase shallow granular flows over variable topography
Marica Pelanti, François Bouchut and Anne Mangeney
ESAIM: M2AN 42 (2008) 851-885
DOI: http://dx.doi.org/10.1051/m2an:2008029
L2 stability analysis of the central discontinuous Galerkin method and a comparison between the central and regular discontinuous Galerkin methods
ESAIM: M2AN has started a new practice of occasionally selecting certain papers as "Highlight" papers. The papers so designated we believe will be of particular interest to our readership due to the topic or approach pursued or perhaps the reach and implications of the results. Highlight papers will be briefly introduced by an Editor and will be available (without charge, and indefinitely) on the ESAIM: M2AN electronic repository.
The present paper combines two methodologies: The discontinuous Galerkin method and the centered scheme method. The discontinuous Galerkin method has incorporated successful features of finite volume schemes for solving hyperbolic PDEs with shocked solutions. The centered scheme framework is a finite volume / finite difference methodology which avoids explicit Riemann solvers. The analysis provided in this paper indicates that there might be advantages over the traditional discontinuous Galerkin method and the potential to use the difference between the duplicative information over staggered meshes to control numerical dissipation and to possibly guide adaptivity.
David Gottlieb, Associate Editor,
Claude Le Bris and Anthony T. Patera, Editors-in-Chief
L2 stability analysis of the central discontinuous Galerkin method and a comparison between the central and regular discontinuous Galerkin methods
Yingjie Liu, Chi-Wang Shu, Eitan Tadmor and Mengping Zhang
ESAIM: M2AN 42 (2008) 593-607
DOI: http://dx.doi.org/10.1051/m2an:2008018
On the motion of a body in thermal equilibrium immersed in a perfect gas
ESAIM: M2AN has started a new practice of occasionally selecting certain papers as "Highlight" papers. The papers so designated we believe will be of particular interest to our readership due to the topic or approach pursued or perhaps the reach and implications of the results. Highlight papers will be briefly introduced by an Editor-in-Chief, and will be available (without charge, and indefinitely) on the ESAIM: M2AN electronic repository.
Understanding the microscopic picture of macroscopic phenomena is a long-term mathematical challenge. Important progress has been made in recent years in several different domains. This Highlight paper addresses the question of the modelling of friction at the microscopic scale: it studies the motion of a body immersed in a gas which experiences viscous friction owing to collisions with the gas molecules. In the case of diffusive collisions, the study establishes the rate at which the velocity of the body approaches its equilibrium value, and relates this rate to the recollisions of the gas molecules with the surface of the body. Such a study exemplifies the crucial role played by sophisticated mathematical arguments in the intimate understanding, and appropriate modelling, of physical issues.
Claude Le Bris, and Anthony T. Patera
Editors-in-Chief
On the motion of a body in thermal equilibrium immersed in a perfect gas
Kazuo Aoki, Guido Cavallaro, Carlo Marchioro and Mario Pulvirenti
ESAIM: M2AN 42 (2008) 263-275
DOI: http://dx.doi.org/10.1051/m2an:2008007
Proper orthogonal decomposition for optimality systems
ESAIM: M2AN has started a new practice of occasionally selecting certain papers as "Highlight" papers. The papers so designated we believe will be of particular interest to our readership due to the topic or approach pursued or perhaps the reach and implications of the results. Highlight papers will be briefly introduced by an Editor, and will be available (without charge, and indefinitely) on the ESAIM: M2AN electronic repository.
The topic of "reduced order modelling" is increasingly important in the many-query and real-time contexts. In particular, in the areas of parameter estimation, design, optimization, and control, classical techniques are often not sufficiently responsive to be of practical value: new approaches such as reduced order modelling will be required. Many theoretical and computational issues remain unresolved within the reduced order modelling framework, from optimal and efficient sampling to effective primal-dual approaches to treatment of nonlinearities and discontinuities to sharp a priori and a posteriori error estimation. This Highlight Paper addresses one of the concerns crucial to ultimate adoption of reduced order modelling concepts in the optimization and control frameworks: how can reduced order models be constructed that reflect - and are accurate within - the parts of the parameter or "dynamical" space relevant to the desired objectives and constraints?
Claude Le Bris, and Anthony T. Patera
Editors-in-Chief
Proper orthogonal decomposition for optimality systems
Karl Kunisch and Stefan Volkwein
M2AN 42 (2008) 1-23
DOI: http://dx.doi.org/10.1051/m2an:2007054