In an earlier paper, we studied the approximation of solutions V(t) to a class of SPDEs by the empirical measure Vⁿ(t) of a system of n interacting diffusions. In the present paper, we consider a central limit type problem, showing that \sqrt{n}(Vⁿ-V )n converges weakly, in the dual of a nuclear space, to the unique solution of a stochastic evolution equation. Analogous results in which the di.usions that determine Vⁿ are replaced by their Euler approximations are also discussed.

We derive new models for gravity driven shallow water flows in several space dimensions over a general topography. A first model is valid for small slope variation, i.e. small curvature, and a second model is valid for arbitrary topography. In both cases no particular assumption is made on the velocity profile in the material layer. The models are written for an arbitrary coordinate system, and several formulations are provided. A Coulomb friction term is derived within the same framework, relevant in particular for debris avalanches. All our models are invariant under rotation, admit a conservative energy equation, and preserve the steady state of a lake at rest.

In this paper, we design a domain decomposition algorithm for the two-dimensional parabolic equation with variable coefficient by using a larger spacing at interface points and the implicit scheme at the interior points, hence get an algorithm with the relaxed stability bounds. Then we prove the stability and analyze the accuracy of the algorithm by using the idea of maximum principle. Some results of numerical experiments are also provided.

We consider in this article the damped and driven two-dimensional Navier-Stokes equations at the limit of small viscosity coefficient v --> 0⁺. In particular, we obtain upper bounds of the order v^-1 on the fractal and Hausdorff dimensions of the global attractor for the system on the torus T², on the sphere S² and in a bounded domain. Furthermore, in the case of the torus, we establish a lower bound of the order v^-1. This sharp estimate is remarkably smaller than the well established sharp bound for the dimension of the global attractor of the Navier-Stokes equations on the torus T², which is of the order v^-4/3. This means that the damping/friction term plays a significant role in reducing the number of degrees of freedom in this two-dimensional model. This, we believe, is done by dissipating the energy at the large spatial scales which is transferred to these scales via the inverse cascade mechanism. Finally, we remark that the system of equations studied here is related to the Stommel-Charney barotropic ocean circulation model of the gulf stream.

We study the dynamics of two-layer, stratified shallow water flows. This is a model in which two scenarios for eventual mixing of stratified flows (shear-instability and internal breaking waves) are, in principle, possible. We find that unforced flows cannot reach the threshold of shear-instability, at least without breaking first. This is a fully nonlinear stability result for a model of stratified, sheared flow. Mathematically, for 2X2 autonomous systems of mixed type, a criterium is found deciding whether the elliptic domain is reachable -smoothly- from hyperbolic initial conditions. If the characteristic fields depend smoothly on the system's Riemann invariants, then the elliptic domain is unattainable. Otherwise, there are hyperbolic initial conditions that will lead to incursions into the elliptic domain, and the development of the associated instability.

In this paper we are concerned with the limiting behavior of gas flow in a thin channel as described by the Broadwell model. The Broadwell model is a simplified kinetic description for gas dynamics where the main assumption is that the particle distribution function can be represented by a discrete number of velocities. Starting from the Broadwell model and the appropriate boundary conditions we derive two 1D models for gas transport in a thin channel. In the limit of no interparticle collisions the 1D model is the well known telegraph equation. In the case of collisional flow the 1D model is a system of three first-order hyperbolic PDEs. Both 1D models are validated through numerical simulations that compare the 1D models to the 2D Broadwell system. Furthermore, in the limit of no inter-particle collisions we are able to rigorously show that under a di.usive scaling the solutions of the full Broadwell model converge weakly to solutions of the diffusion equation. Under a hyperbolic scaling we are able to show that solutions to the collisionless Broadwell model converge weakly to the solutions of the telegraph equation. Finally, we derive a long-time asymptotic formula for the solution of the collisionless Broadwell system, which reveals oscillations that explain why the convergence in the di.usive and hyperbolic scalings must be weak. Due to the nonlinearity of the inter-particle collisions, we are not able to prove rigorous convergence results for the collisional Broadwell system.

In this paper we study the large time step (LTS) Godunov scheme for scalar hyperbolic conservation laws proposed by LeVeque. We show that for an arbitrary Courant number, all the possible wave interactions in each time step occur only in a finite number of cells, and the number of cells is bounded by a constant depending on the Courant number for a given initial value problem. As an application of the result mentioned above, we show that for any given Courant number, if the total variation of the initial value satisfies some conditions, then the numerical solutions of the LTS Godunov scheme will converge to the entropy solutions of the convex scalar conservation laws.

An algorithm is presented for local space-time mesh refinement appropriate for electromagnetic simulations based on the space-time staggered FDTD method. The method is based on the adaptive mesh re.nement algorithm originally developed for hyperbolic conservation laws. Analysis of the dispersion relation and of the numerical reflection and transmission coefficients in one and two space dimensions shows that a scheme based on linear interpolation at the grid interfaces is unstable due to reflection coefficient > 1 at frequencies above the cutoff frequency of the coarse grid. A second-order accurate algorithm based on higher-order interpolations that enforces conservation of the magnetic field circulation at the fine-coarse grid boundaries is constructed. The new algorithm is shown to be stable and accurate for long time integration. A numerical simulation of an optical ring microcavity resonator using multilevel grid refinement in two space dimensions is presented.

We consider the Liouville equations with highly heterogeneous Hamiltonians and their numerical solution by a time splitting algorithm. Such equations model the density of particles evolving according to the corresponding Hamiltonian dynamics as well as the propagation of high frequency waves with a wavelength much smaller than the correlation length of the random Hamiltonian. Our main results are on the relation between the time step used in the time splitting algorithm and the correlation length of the Hamiltonian. In order to fully resolve the Liouville equation, the time step must be chosen much smaller than the correlation length. However, we show that the time step can be chosen of the same order as the correlation length of the Hamiltonian when one is only interested in suitable statistical properties of the solution to the Liouville equation. We also present a more involved time splitting algorithm that allows us to take a time step independent of the correlation length.