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.

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.

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-Bénard 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-Bénard 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].

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.

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.

We present a multiple-patch phase space method for computing trajectories on two-dimensional manifolds possibly embedded in a higher-dimensional space. The dynamics of trajectories are given by systems of ordinary differential equations (ODEs). We split the manifold into multiple patches where each patch has a well-defined regular parameterization. The ODEs are formulated as escape equations, which are hyperbolic partial differential equations (PDEs) in a three-dimensional phase space. The escape equations are solved in each patch, individually. The solutions of individual patches are then connected using suitable inter-patch boundary conditions. Properties for particular families of trajectories are obtained through a fast post-processing. We apply the method to two different problems: the creeping ray contribution to mono-static radar cross section computations and the multivalued travel-time of seismic waves in multi-layered media. We present numerical examples to illustrate the accuracy and efficiency of the method.

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.

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.

We consider the continuum limit for models of solids that arise in density functional theory and the QM-continuum approximation of such models. Two different versions of QM- continuum approximation are proposed, depending on the level at which the Cauchy-Born rule is used, one at the level of electron density and one at the level of energy. Consistency at the interface between the smooth and the non-smooth regions is analyzed. We show that if the Cauchy-Born rule is used at the level of electron density, then the resulting QM-continuum model is free of the so-called “ghost force” at the interface. We also present dynamic models that bridge naturally the Car-Parrinello method and the QM-continuum approximation.

This paper is concerned with the derivation of uniform bounds with respect to the scaled Planck constant ${\cal E}$ for solutions to the open transient Schrödinger-Poisson system introduced by Ben Abdallah et al in [Math.Meth.Mod. in App. Sci., 15, 667-688, 2005]. The uniform estimates stem from a careful analysis of the non-local in time transparent boundary conditions which allow to restrict the original problem posed on an unbounded domain to a bounded domain of interest. These bounds can be used to obtain the semi-classical limit of the system. The paper also gives an existence and uniqueness result for weak solutions while they were previously defined in a strong sense.

When one is attempting to build a general boundary value problem solver which uses Green’s functions, it is often desirable to automatically transform inhomogeneous boundary value problems into homogeneous boundary value problems. If the sum of the boundary matrices is singular, a change of coordinates is required which can transform the inhomogeneous boundary value problem into a homogeneous boundary value problem. This change of coordinates is an invertible C1 path in matrix space with endpoints that can multiply the boundary matrices to produce a nonsingular sum. Here we propose a simple automatic coordinate transformation that performs this task. We prove that this algorithm constructs a coordinate transformation whenever such a transformation exists, and provide numerical examples illustrating the practicability of the method.

We consider the numerical solution of the linear Klein-Gordon equation in $\bbfR^2\times$ and $\bbfR^3\times$. An artificial boundary is introduced to obtain a bounded computational domain. On the given artificial boundary, the exact boundary condition and a series of approximating boundary conditions are constructed, which are called absorbing boundary conditions. By using either the exact or approximating boundary conditions on the artificial boundary, the original problem is reduced to either an equivalent or an approximately equivalent initial-boundary value problem on the bounded computational domain. The uniqueness of the approximate problem is then proved. The numerical results demonstrate that the method given in this paper is effective and feasible.