Naoufel Ben Abdallah: WKB Schemes for the Schrodinger equation
The Schrodinger equation is one of the most used models for the simulation of quatum transport in electronic nanostructures. Macroscopic quantities such as particle density or current density are computed as an integral over the energy variable of single state quantities.
Numerically, the integral is computed thanks to a suitable numerical integration method and implies a large number of Schrodinger equations to be solved. An energy grid containing a certain amount of points is constructed and the wavefunction for each of these points is computed by solving the Schroedinger equation. For high energies, the single states have a small de Broglie length and oscillate at much smaller space scale than for low energies.
Besides, the macroscopic quantities like particle density are relatively smooth functions of the position variable.
Using the same spatial grid for all the energies to solve the Schroedinger equations with standard finite element or finite difference methods requires a large number of points thus increasing unnecessarily the numerical cost. The approach adapted here is to use the WKB asysmptotic in order to reduce the number of grid points. Indeed, the need for a refined spatial grid is due to the linear or polynomial interpolation underlying the standard finite difference or finite element methods. Therefore, if the oscillation phase is known accurately, the phase factor could be used to interpolate the nodal values of the wave function and a coarser grid can be allowed. In the one dimensional case, this can be done since the WKB asymptotics provide us with an explicit formula for this phase factor. We shall also present the waveguide case where the multidimensional Schroedinger equation can be written as a copupled system of one dimensional Schrodinger equation and for which an approximate WKB approximation is used.

Yann Brenier: L2 analysis of some adhesion models
Adhesion dynamics, involving particles, springs or strings, can be modelled by hyperbolic conservation laws with concentration effects, which yields challenging numerical and analysis problems. Typical examples are pressureless or Chaplygin gas equations.
The analysis of conservation laws and Hamilton-Jacobi equations developed by Kruzhkov, Volpert, Crandall-Lions and many others in the 70's and 80's involves semi-groups on non reflexive spaces, typically L1 and C0. We show that, at least in one space dimension, using lagrangian coordinates, a parallel and equivalent analysis can be performed in L2. At the numerical level, we introduce very efficient schemes in one space dimension and discuss their multidimensional extension by dimensional splitting and eulerian/lagrangian projections.

Pierre Degond: Quantum drift-diffusion models derived from an entropy minimization principle (joint work with S. Gallego, F. Mehats and C. Ringhofer)
In this work, we give an overview of recently derived quantum hydrodynamic and diffusion models. A quantum local equilibrium is defined as a minimizer of the quantum entropy subject to local moment constraints (such as given local mass, momentum and energy densities). These equilibria relate the thermodynamic parameters (such as the temperature or chemical potential) to the densities in a non-local way.
Quantum hydrodynamic models are obtained through moment expansions of the quantum kinetic equations closed by quantum equilibria.
We also derive collision operators for quantum kinetic models which decrease the quantum entropy and relax towards quantum equilibria. Then, through diffusion limits of the quantum kinetic equation, we establish various classes of models which are quantum extensions of the classical energy-transport and drift-diffusion models. We shall present 1D numerical results which show that these model are able to reproduce the major features of quantum transport in resonant tunelling structures.

Francis Filbet: Spectral methods for the collisional motion of (heated) granular flows
In this talk, I will present a spectral method for inelastic Boltzmann equation describing the collisional motion of a ganular gas with and without a heating source. The schemes are based on a Fourier representation of the equation in velocity space and provide a very accurate description of the time evolution of the distribution function. Several numerical results in dimension one and three show the efficiency and accuracy of the proposed method. Some mathematical and physical conjectures are also addresses with the aid of numerical simulations.

David Levermore: Transition Regime Models from Linear Kinetic Equations
A framework is presented for building transition regime models from linear kinetic equations. This framework combines elements of two traditional approaches to this problem: moment closures and Chapman-Enskog expansions. The framework has three components: a stationary balance temporal approximation, a small-gradient expansion as an interior spatial approximation, and a natural framework for developing boundary conditions. The resulting models will be formally well-posed, capture the correct stationary asymptotics, and properly dissipate. We present a variety of such transition regime models in the context of monoenergetic, photon transport through a stationary, isotropic medium that scatters, absorbs, and emits as a blackbody.

Pierre Louis Lions: Generalized stochastic flows and kinetic models for polymer flows
We present some kinetic models for polymer flows. These models involve a coupling of Navier-Stokes and Fokker Plank equations (or other form of stochastic differential equations).
We discuss in this talk the existence of global weak solutions (joint work with N. Masmoudi) and one of the used tools, namely, generalized stochastic flows (joint work with C. Lebris)

Hailiang Liu: Computing multi-valued physical observables for the high frequency limit of symmetric hyperbolic systems
We develop a level set method for the computation of multi-valued physical observables (density, velocity, energy, etc.) for the high frequency limit of symmetric hyperbolic systems in any space dimensions. We take two appraoches to derive the method.
The first one starts with a decoupled system of eiconal for phase $S$ and transport equations for density $\rho$: The main idea is to evolve the density near the $n$-dimensional bi-characteristic manifold of the eiconal (Hamiltonian-Jacobi) equation, that is identified as the common zeros of $n$ level set functions in phase space. These level set functions are generated from solving the Liouville equation with initial data chosen to embed the phase gradient. Simultaneously we track a new quantity $f$ by solving again the Liouville equation near the obtained zero level set but with initial density as initial data. The multi-valued density and higher moments are thus resolved by integrating $f$ along the bi-characteristic manifold in the phase directions.
The second one uses the high frequency limit of symmetric hyperbolic systems derived by the Wigner transform. This gives rise to Liouville equations in the phase space with measure-valued solution in its initial data. Thanks to the linearity of the Liouville equation we can decompose the density distribution into products of functions each of which solves the Liouville equations with $L^\infty$ initial data on any bounded domain. It yields higher order moments such as energy and energy flux.
The main advantages of this new approach, in contrast to the standard kinetic equation approach using the Liouville equation with a Dirac measure initial data, include: 1) the Liouville equations are solved with $L^{\infty}$ initial data, and a singular integral involving the Dirac-$\delta$ function is evaluated only in the post-processing step, thus avoiding oscillations and excessive numerical smearing; 2) a local level set method can be utilized to significantly reduce the computation in the phase space. These advantages allow us to compute {\it all} physical observables for multidimensional problems.
Our method applies to the wave fields corresponding to simple eigenvalues of the dispersion matrix. One such example is the wave equation, which will be studied numerically in this paper.
This is a joint work with S. Jin, S. Osher and R. Tsai.

Peter Markowich: On Asymptotic Regimes for the Maxwell-Dirac System
We present the Maxwell-Dirac system modelling the quantum-relativistic transport of fast spin 1/2 particles. In particular we shall focus on numerical techniques and simulations in two asymptotic regimes:
1)the semiclassical limit
2)the non-relativistic limit.

Christian Ringhofer: Cancelled.

Sergej Rjasanow: Stochastic numerics for the Boltzmann equation.
Abstract

Jin Shi: Hamiltonian-preserving schemes for the Liouville equation with discontinuous potentials
When numerically solving the Liouville equation with a discontinuous potential, one faces the problem of zero time step due to the CFL constraint, and the inconsistency to the constant Hamiltonian. In this paper, we propose a class of Hamiltonian-preserving schemes that are able to overcome these numerical deficiencies. The key idea is to build into the numerical flux the behavior of a classical particle at a potential barrier. We establish the stability theory of these new schemes, and analyze their numerical accuracy. Numerical experiments are carried out to verify the theoretical results.
This method can also be applied to the level set methods for the computations of multivalued physical observables in the semiclassical limit of the linear Schrodinger equation with a discontinuous potential, among other applications.

Giovanni Russo: Computation of Strained Epitaxial Growth in Three Dimensions by Kinetic Monte Carlo.
A numerical method for computation of heteroepitaxial growth in the presence of strain is presented. The model used is based on a solid-on-solid model with a cubic lattice. Elastic effects are incorporated using a ball and spring type model. The growing film is evolved using Kinetic Monte Carlo (KMC) and it is assumed that the film is in mechanical equilibrium. The strain field in the substrate is computed by an exact solution which is efficiently evaluated using the fast Fourier transform. The strain field in the growing film is computed directly. The resulting coupled system is solved iteratively using the conjugate gradient method. Finally we introduce various approximations in the implementation of KMC to improve the computation speed. Numerical results show that layer-by-layer growth is unstable if the misfit is large enough resulting in the formation of three dimensional islands. Further development using multigrid approach will be addressed.

Eitan Tadmor: Semi Classical Limits with Sub-Critical Initial Data
We discuss the semi-classical limit of nonlinear Schrödinger-Poisson (NLSP) equation which is realized in terms of the density-velocity pair, governed by the Euler-Poisson equation. We show that the Euler-Poisson and related equations admit global smooth solutions subject to initial data below certain critical thresholds; consequently, we can justify the NLSP semi-classical limit in such sub-critical regimes.

Chi Wang Shu: High Order WENO and Discontinuous Galerkin Methods for Transport Problems
In this talk we will describe the high order finite difference WENO schemes and discontinuous Galerkin methods for solving transport problems. General algorithm issues, comparison of the two algorithms, and applications to semiconductor device simulations and nonlinear dispersive wave equations will be addressed.

Eric Vanden-Eijnden: Multiscale Kinetic Monte-Carlo Scheme with Application to Chemical Reacting Systems
An efficient computational strategy will be presented for Kinetic Monte-Carlo (KMC) schemes with multiple time-scales. These types of KMC arise e.g. in the modeling of spatially homogeneous or well-stirred chemical systems with species reacting at very different rates. The multiscale strategy is based on averaging theorems for continuous-in-time Markov chains, which allow one to identify groups of slow and fast variables and simulate the evolution of each on its natural time-scale, thereby resulting in substantial efficiency gain in computational cost. This is a joint work with Weinan E and Di Liu.

Issues on computational transport in meso and nano scales,

Last modified: 11 February 2005.