Showing papers in "Communications in Mathematical Sciences in 2007"

A second-order well-balanced positivity preserving central-upwind scheme for the saint-venant system ∗

Abstract: A family of Godunov-type central-upwind schemes for the Saint-Venant system of shallow water equations has been first introduced in (A. Kurganov and D. Levy, M2AN Math. Model. Numer. Anal., 36, 397-425, 2002). Depending on the reconstruction step, the second-order versions of the schemes there could be made either well-balanced or positivity preserving, but fail to satisfy both properties simultaneously. Here, we introduce an improved second-order central-upwind scheme which, unlike its forerun- ners, is capable to both preserve stationary steady states (lake at rest) and to guarantee the positivity of the computed fluid depth. Another novel property of the proposed scheme is its applicability to models with discontinuous bottom topography. We demonstrate these features of the new scheme, as well as its high resolution and robustness in a number of one- and two-dimensional examples.

On the well-posedness of the linear peridynamic model and its convergence towards the Navier equation of linear elasticity

Abstract: The non-local peridynamic theory describes the displacement field of a continuous body by the initial-value problem for an integro-differential equation that does not include any spatial derivative. The non-locality is determined by the so-called peridynamic horizon $\delta$ which is the radius of interaction between material points taken into account. Well-posedness and structural properties of the peridynamic equation of motion are established for the linear case corresponding to small relative displacements. Moreover the limit behavior as $\delta \rightarrow 0$ is studied.

Stability estimates for a class of Helmholtz problems

Abstract: This paper presents new stability estimates for the scalar Helmholtz equation with a complex-valued Robin boundary condition as well as Dirichlet and Neumann boundary conditions. For each estimate, we state the explicit dependency of constants on the wave number. To deal with mixed boundary conditions, we impose geometrical constraints on the two-dimensional or three-dimensional bounded domain.

Global existence for a 2D incompressible viscoelastic model with small strain

Abstract: In this paper, we continue our previous study towards understanding the twodimensional hydrodynamic systems describing Oldroyd type incompressible viscoelastic fluids. We will decompose the deformation tensor into the strain and rotation components and look at their distinct contributions and structures in the small strain (with respect to viscosity) dynamics. In particular, we prove that there exist classical solutions globally in time if the strain component of the initial deformation is small enough, while we require no assumptions on smallness of the magnitude of the rotation component.

Well-posedness of 3D vortex sheets with surface tension

Abstract: We prove well-posedness for the initial value problem for a vortex sheet in 3D fluids, in the presence of surface tension. We first reformulate the problem by making a favorable choice of variables and parameterizations. We then perform energy estimates for the evolution equations. It is important to note that the Kelvin-Helmholtz instability is present for the vortex sheet in the absence of surface tension. Accordingly, we must construct the energy functional carefully with an eye toward the regularization of this instability. Well-posedness follows from the estimates.

The Riemann problem for the shallow water equations with discontinuous topography

Abstract: We construct the solution of the Riemann problem for the shallow water equations with discontinuous topography. The system under consideration is non-strictly hyperbolic and does not admit a fully conservative form, and we establish the existence of two-parameter wave sets, rather than wave curves. The selection of admissible waves is particularly challenging. Our construction is fully explicit, and leads to formulas that can be implemented numerically for the approximation of the general initial-value problem.

New entropy estimates for the Oldroyd-B model and related models

Abstract: This short note presents the derivation of a new {\it a priori} estimate for the Oldroyd-B model. Such an estimate may provide useful information when investigating the long-time behaviour of macro-macro models, and the stability of numerical schemes. We show how this estimate can be used as a guideline to derive new estimates for other macroscopic models, like the FENE-P model.

Analysis of optimal boundary control for radiative heat transfer modeled by the $SP_{n}$-system

Abstract: We present an analytic study of an optimal boundary control problem for the diffusive $SP_{1}$-system modeling radiative heat transfer. The cost functional is of tracking-type and the control problem is considered as a constrained optimization problem, where the constraint is given by the nonlinear parabolic/elliptic $SP_{1}$-system. We prove the existence, uniqueness and regularity of bounded states, which allows for the introduction of the reduced cost functional. Further, we show the existence of an optimal control, derive the first-order optimality system and analyze the adjoint system, for which we prove existence, uniqueness and regularity of adjoint states.

Multiscale hierarchical decomposition of images with applications to deblurring, denoising and segmentation ∗

Abstract: We extend the ideas introduced in (33) for hierarchical multiscale decompositions of images. Viewed as a function f ∈ L2(­), a given image is hierarchically decomposed into the sum or product of simpler "atoms" uk, where uk extracts more refined information from the previous scale uki1. To this end, the uk's are obtained as dyadically scaled minimizers of standard functionals arising in image analysis. Thus, starting with vi1 := f and letting vk denote the residual at a given dyadic scale, ¸k ∼ 2k, the recursive step (uk,vk) = arginf QT (vki1,¸k) leads to the desired hierarchical decomposition, f ∼ P T uk; here T is a blurring operator. We characterize such QT -minimizers (by duality) and expand our previous energy estimates of the data f in terms of k ukk . Numerical results illustrate applications of the new hierarchical multiscale decomposition for blurry images, images with additive and multiplicative noise and image segmentation.

Empirical evaluation of a sub-linear time sparse DFT algorithm

Abstract: In this paper we empirically evaluate a recently proposed Fast Approximate Discrete Fourier Transform (FADFT) algorithm, FADFT-2, for the first time. FADFT-2 returns approximate Fourier representations for frequency-sparse signals and works by random sampling. Its implemen- tation is benchmarked against two competing methods. The first is the popular exact FFT imple- mentation FFTW Version 3.1. The second is an implementation of FADFT-2’s ancestor, FADFT-1. Experiments verify the theoretical runtimes of both FADFT-1 and FADFT-2. In doing so it is shown that FADFT-2 not only generally outperforms FADFT-1 on all but the sparsest signals, but is also significantly faster than FFTW 3.1 on large sparse signals. Furthermore, it is demonstrated that FADFT-2 is indistinguishable from FADFT-1 in terms of noise tolerance despite FADFT-2’s better execution time.

KPP pulsating front speed-up by flows

Abstract: We obtain a criterion for pulsating front speed-up by general periodic incompressible flows in two dimensions and in the presence of KPP nonlinearities. We achieve this by showing that the ratio of the minimal front speed and the effective diffusivity of the flow is bounded away from zero and infinity by constants independent of the flow. We also study speed-up of reaction-diffusion fronts by various examples of flows in two and three dimensions.

Rayleigh Bénard convection: dynamics and structure in the physical space

Abstract: The main objective of this article is part of a research program to link the dynamics of fluid flows with the structure of these fluid flows in physical space and the transitions of this structure. To demonstrate the main ideas, we study the two-dimensional Rayleigh-Benard convection, which serves as a prototype problem. The analysis is based on two recently developed nonlinear theories: geometric theory for incompressible flows [T. Ma and S. Wang, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 119, 2005] and bifurcation and stability theory for nonlinear dynamical systems (both finite and infinite dimensional) [T. Ma and S. Wang, World Scientific, 2005]. We have shown in [T. Ma and S. Wang, Commun. Math. Sci., 2(2), 159–183, 2004] that the Rayleigh-Benard problem bifurcates from the basic state to an attractor AR when the Rayleigh number R crosses the first critical Rayleigh number Rc for all physically sound boundary conditions, regardless of the multiplicity of the eigenvalue Rc for the linear problem. In this article, in addition to a classification of the bifurcated attractor AR, the structure of the solutions in physical space and the transitions of this structure are classified, leading to the existence and stability of two different flows structures: pure rolls and rolls separated by a cross the channel flow. It appears that the structure with rolls separated by a cross-channel flow has not been carefully examined although it has been observed in other physical contexts such as the Branstator-Kushnir waves in atmospheric dynamics [G.W. Branstator, J. Atmos. Sci., 44, 2310–2323, 1987] and [K. Kushnir, J. Atmos. Sci., 44, 2727–2742, 1987].

Interpolation between Logarithmic Sobolev and Poincare Inequalities

Abstract: This note is concerned with intermediate inequalities which interpolate between the logarithmic Sobolev and the Poincare inequalities. For such generalized Poincare inequalities we improve upon the known constants from the literature. Cette note est consacree a des inegalites intermediaires qui interpolent entre les inegalites de Sobolev logarithmiques et les inegalites de Poincare. Pour de telles inegalites de Poincare generalisees, nous ameliorons les constantes donnees dans la litterature.

ON HMM-like integrators and projective integration methods for systems with multiple time scales

Abstract: HMM-like multiscale integrators and projective integration methods are two different types of multiscale integrators which have been introduced to simulate efficiently systems with widely disparate time scales. The original philosophies of these methods, reviewed here, were quite different. Recently, however, projective integration methods seem to have evolved in a way that make them increasingly similar to HMM-integrators and quite different from what they were originally. Nevertheless, the strategy of extrapolation which was at the core of the original projective integration methods has its value and should be extended rather than abandoned. An attempt in this direction is made here and it is shown how the strategy of extrapolation can be generalized to stochastic dynamical systems with multiple time scales, in a way reminiscent of Chorin’s artificial compressibility method and the Car-Parrinello method used in molecular dynamics. The result is a seamless integration scheme, i.e. one that does not require knowing explicitly what the slow and fast variables are.

A short review on the derivation of the nonlinear quantum Boltzmann equations

Abstract: In this review paper we describe the problem of deriving a Boltzmann equation for a system of N interacting quantum particles, under the appropriate scaling limits. We mainly follow the approach developed by the authors in previous works. From a rigorous viewpoint, only partial results are available, even for short times, so that the complete problem is still open. 1. Introduction A large quantum system of N identical interacting particles can often be described in terms of a Boltzmann equation. This is an asymptotic model: the equation given from flrst principles is the N body Schrodinger equation. As such, the Boltzmann description only holds in suitable regimes, namely when the number of particles is large, and when the interaction potential between pairs of particles has a small efiect. Concerning this last point, two quite difierent settings are relevant. In the so-called weak-coupling limit, the interaction potential itself is small, while the gas is dense: the typical distance between particles is of order one. In the low-density regime at variance, the elementary interaction potential is of order one, while the gas is rarefled: the typical distance between particles is large, hence the efiect of the pairwise interactions is small.

Construction and validation of a rigorous surface hopping algorithm for conical crossings

Abstract: This article presents and evaluates a surface hopping algorithm for time-dependent two-level Schrodinger systems with conically intersecting eigenvalues. The algorithm implements an asymptotic semigroup for approximating the solution's Wigner function, which was rigorously defined and derived from the Schrodinger equation by two of the authors in previous work. It is applied to two-dimensional isotropic systems, which include linear Jahn-Teller Hamiltonians and Gaussian initial data. It reproduces energy level populations and expectation values with an accuracy of two to three percent.

Transport-equilibrium schemes for computing contact discontinuities in traffic flow modeling

Abstract: We present a very efficient numerical strategy for computing contact discontinuities in traffic flow modeling. We consider the Aw-Rascle model, and the objective is to remove spurious oscillations generated for instance by the Godunov method near contact discontinuities. The method is mixed and based on both a random sampling strategy and the Godunov method. To prove the validity of the method, we show that it enjoys important stability properties and propose numerical tests. The convergence of the algorithm is demonstrated numerically.

Physical and numerical viscosity for quantum hydrodynamics

Abstract: Viscous stabilizations of the quantum hydrodynamic equations are studied. The quantum hydrodynamic model consists of the conservation laws for the particle density, momen- tum, and energy density, including quantum corrections from the Bohm potential. Two different stabilizations are analyzed. First, viscous terms are derived using a Fokker-Planck collision operator in the Wigner equation. The existence of solutions (with strictly positive particle density) to the isothermal, stationary, one-dimensional viscous model for general data and nonhomogeneous boundary conditions is shown. The estimates depend on the viscosity and do not allow to perform the inviscid limit. Second, the numerical viscosity of the second upwind finite-difference discretization of the inviscid quantum hydrodynamic model is computed. Finally, numerical simulations using the non-isothermal, stationary, one-dimensional model of a resonant tunnelling diode show the influence of the viscosity on the solution.

Mathematical Analysis of Multi-Scale Models of Complex Fluids

Abstract: The state of the art of the mathematical and numerical analysis of multi-scale models of complex fluids is reviewed. Issues addressed include well-posedness of the models, convergence analysis of the numerical methods, and the structure of stationary solutions of the Doi-Onsager equation.

On quantum hydrodynamic and quantum energy transport models

Abstract: In this paper, we consider two recently derived models: the Quantum Hydrodynamic model (QHD) and the Quantum Energy Transport model (QET). We propose different equivalent formulations of these models and we use a commutator formula for stating new properties of the models. A gauge invariance lemma permits to simplify the QHD model for irrotational flows. We finish by considering the special case of a slowly varying temperature and we discuss possible approximations which will be helpful for future numerical discretizations.

Mass and energy balance laws derived from High-Field limits of thermostatted Boltzmann equations

Abstract: We derive coupled mass and energy balance laws from a High-Field limit of thermostatted Boltzmann equations. The starting point is a Boltzmann equation for elastic collisions subjected to a large force field. By adding a thermostat correction, it is possible to expand the solutions about a High-Field equilibrium obtained when balancing the thermostatted field drift operator with the elastic collision operator. To this aim, a hydrodynamic type scaling of the thermostatted Boltzmann equation is used, considering that the leading 'collision operator' actually consists of the combination of the thermostatted field operator and of the elastic collision operator. At leading order in the Knudsen number, the resulting model consist of coupled nonlinear first order partial differential equations. We investigate two cases. The first one is based on a one-dimensional BGK-type operator. The second one is three dimensional and concerns a Fokker-Planck collision operator. In both cases, we show that the resulting models are hyperbolic, thereby indicating that they might be appropriate for a use in physically realistic situations.

The TDHF Approximation for Hamiltonians with $m$-particle Interaction Potentials

Abstract: According to a theory of H. Spohn, the time-dependent Hartree (TDH) equation governs the 1-particle state in N -particle systems whose dynamics are prescribed by a non-relativistic Schrodinger equation with 2-particle interactions, in the limit N tends to infinity while the strength of the 2-particle interaction potential is scaled by 1/N . In previous work we have considered the same mean field scaling for systems of fermions, and established that the error of the time-dependent Hartree-Fock (TDHF) approximation tends to 0 as N tends to infinity. In this article we extend our results to systems of fermions with m-particle interactions (m > 2). 1 The TDHF equation as a mean field approximation The time-dependent Hartree Fock (TDHF) equation [1] is an attempt to approximate the state of a system of interacting fermions by one time-dependent Slater determinant (thus discarding any “correlation” in the many electron system, cf. [4]). In our papers [2, 3] we have derived the TDHF dynamics as that of a single fermion in the mean field, in the spirit of Spohn’s derivation of the time-dependent Hartree equation [5] and refinements thereof [6, 7, 8] (see [9] for a good overview). Here we show how the theorem of [2] for 2-particle interactions may be generalized to cases where the N -particle Hamiltonian involves m-particle interactions with m > 2. Let H be a Hilbert space and let Hn denote the n tensor power of H, i.e., HN = n times { }} { H⊗ H⊗ · · · ⊗ H . For π in the group Sn of permutations of {1, 2, . . . , n}, define the unitary “permutation” operator Uπ by Uπ(x1 ⊗ ...⊗ xn) = xπ−1(1) ⊗ ...⊗ xπ−1(n) for all x1, . . . , xn ∈ H. Define An = ∑ π∈Sn sgn(π)Uπ (1) for all n ∈ N. Then 1 n!An is an orthogonal projector whose range is the space of antisymmetric vectors in Hn. Consider N identical fermions whose 1-particle Hilbert space is H. The appropriate N -fermion Hilbert space is the space of antisymmetric wavefunctions in HN , i.e., the range of the orthogonal projector 1 N !AN . If {ej}j∈J is an orthonormal basis of H then the set { 1 √ N ! AN (ej1 ⊗ ej2 ⊗ · · · ⊗ ejN ) : {j1, j2, . . . , jN} ⊂ J } ∗Univ Paris 7 and Laboratoire J.L. Lions (Univ. Paris 6), France (bardos@math.jussieu.fr). †CEA/DAM Ile de France, DPTA/Service de Physique Nucleaire, BP 12, F-91680 Bruyeres-le-Châtel, France (bernard.ducomet@cea.fr). ‡Laboratoire J.L. Lions (Univ. Paris 6), France (golse@math.jussieu.fr). §Wolfgang Pauli Inst., Nordbergstr. 15, A–1090 Wien, Austria (alex@alexgottlieb.com). ¶WPI c/o Fak. f. Math., Univ. Wien, Nordbergstr. 15, A–1090 Wien, Austria (mauser@courant.nyu.edu).

Homogenization for inertial particles in a random flow

Abstract: We study the problem of homogenization for inertial particles moving in a time-dependent random velocity field and subject to molecular diffusion. We show that, under appropriate assumptions on the velocity field, the large-scale, long-time behavior of the inertial particles is governed by an effective diffusion equation for the position variable alone. This is achieved by the use of a formal multiple scales expansion in the scale parameter. The expansion relies on the hypoellipticity of the underlying diffusion. An expression for the diffusivity tensor is found and various of its properties are studied. The results of the formal multiscale analysis are justified rigorously by the use of the martingale central limit theorem. Our theoretical findings are supported by numerical investigations where we study the parametric dependence of the effective diffusivity on the various non-dimensional parameters of the problem.

Seamless multiscale modeling of complex fluids using fiber bundle dynamics

Abstract: We present a seamless multiscale model and an efficient coupling scheme for the study of complex fluids. The multiscale model consists of macroscale conservation laws for mass and momentum, molecular dynamics on fiber bundles, as well as the Irving-Kirkwood formula which links the macroscale stress tensor with the microscopic variables. The macroscale and microscale models are solved with a macro time step and a micro time step respectively. The two models are synchronized at each time step by exchanging the velocity gradient (from macro to micro) and the stress tensor (from micro to macro). The multiscale method is applied to study the dynamics of polymer fluids in a channel driven by external forces.

Diffuse interface energies capturing the euler number: relaxation and renomalization ∗

Abstract: We introduce a set of new interfacial energies for approximating the Euler number of level surfaces in the phase field (diffuse-interface) representation. These new formulae have simpler forms than those studied earlier in (Q. Du, C. Liu and X. Wang, Retrieving topological information for phase field models, SIAM J. Appl. Math., 65, 1913-1932, 2005), and do not contain higher order derivatives of the phase field function. Theoretical justifications are provided via formal asymptotic analysis, and practical validations are performed through numerical experiments. Relaxation and renormalization schemes are also developed to improve the robustness of the new energy functionals.

A sub-linear scaling algorithm for computing the electronic structure of materials

Abstract: We introduce a class of sub-linear scaling algorithms for analyzing the electronic structure of crystalline solids with isolated defects. We divide the localized orbitals of the electrons into two sets: one set associated with the atoms in the region where the deformation of the material is smooth (smooth region), and the other set associated with the atoms around the defects (non- smooth region). The orbitals associated with atoms in the smooth region can be approximated accurately using asymptotic analysis. The results can then be used in the original formulation to find the orbitals in the non-smooth region. For orbital-free density functional theory, one can simply partition the electron density into a sum of the density in the smooth region and a density in the non-smooth region. This kind of partition is not used for Kohn-Sham density functional theory and one uses instead the partition of the set of orbitals. As a byproduct, we develop the necessary real space formulations and we present a formulation of the electronic structure problem for a subsystem, when the electronic structure for the remaining part is known.

Seamless multiscale modeling via dynamics on fiber bundles

Abstract: We present a general mathematical framework for modeling the macroscale behavior of a multiscale system using only microscale models, by formulating the effective macroscale models as dynamic models on the underlying fiber bundles. This framework allows us to carry out seamless multiscale modeling using traditional numerical techniques. At the same time, they give rise to an interesting mathematical structure and new interesting mathematical problems. We discuss several examples from homogenization problems, continuum modeling of solids based on atomistic or electronic structure models, macroscale behavior of interacting diffusion, and continuum modeling of complex fluids based on kinetic and Brownian dynamics models.

Weak shock waves for the general discrete velocity model of the Boltzmann equation

Abstract: We study the shock wave problem for the general discrete velocity model (DVM), with an arbitrary finite number of velocities. In this case the discrete Boltzmann equation becomes a system of ordinary differential equations (dynamical system). Then the shock waves can be seen as heteroclinic orbits connecting two singular points (Maxwellians). In this paper we give a constructive proof for the existence of solutions in the case of weak shocks. We assume that a given Maxwellian is approached at infinity, and consider shock speeds close to a typical speed c, corresponding to the sound speed in the continuous case. The existence of a non-negative locally unique (up to a shift in the independent variable) bounded solution is proved by using contraction mapping arguments (after a suitable decomposition of the system). This solution is shown to tend to a Maxwellian at minus infinity. Existence of weak shock wave solutions for DVMs was proved by Bose, Illner and Ukai in 1998. In this paper, we give a constructive proof following a more straightforward way, suiting the discrete case. Our approach is based on earlier results by the authors on the main characteristics (dimensions of corresponding stable, unstable and center manifolds) for singular points to general dynamical systems of the same type as in the shock wave problem for DVMs. The same approach can also be applied for DVMs for mixtures

Quantitative linearized study of the Boltzmann collision operator and applications

Abstract: We present recent results [4, 28, 29] about the quantitative study of the linearized Boltzmann collision operator, and its application to the study of the trend to equilibrium for the spatially homogeneous Boltzmann equation for hard spheres.

Existence of Weak Entropy Solutions for Gas Chromatography Systen with One or Two Active Species and Non Convex Isotherms

Abstract: This paper deals with a system of two equations which describes heatless adsorption of a gaseous mixture with two species. Using the hyperbolicity property of the system with respect to the $(x,t)$ variables, that is with $x$ as the evolution variable, we find all the entropy-flux pairs. Making use of a Godunov-type scheme we obtain an existence result of a weak entropy solution satisfying some BV regularity.