In this paper, we derive the Darwin model in 3-D unbounded domains by the decomposition of the vector field, then we show that the Darwin model approximates the Maxwell's equation up to the second order for the magnetic flux density, and to the third order for the electric field with respect to \eta=\bar{v}/c, where \bar{v} is the characteristic velocity, and c is the speed of light.

We introduce a new simple Eulerian method for treatment of moving boundaries in compressiuble fluid computations. Our approach is based on the extension of the interface tracking method recently introduced in the context of multifluds. The fluid domain is placed in a rectangular computational domain of a fixed size, which is divided into Cartesian cells. At every time moment, there are three types of cells: internal, boundary, and extrenal ones. The numerical solution is evolved in internal cells only. The numerical fluxes at the cells near the boundary are computed using the teachnique from [A. Chertock, S. Karni and A. Kurganov, M2AN Math. Model. Numer. Anal., submitted] combined with a solid wall ghost-cell extrapolation and an interpolation in the phase space. The proposed computational frameworks is general and may be used in conjunction with one's favorite finite-volume method. The robustmess of the new approach is illustrated on a number of one- and two-dimensional numerical examples.

The distribution of a time integral of geometric Brownian motion is not well understood. To price an Asian option and to obtain measures of its dependence on the parameters of time, strike price, and underlying market price, it is essential to have the distribiution of time integral of geometric Brownian motion and it is also required to have a way to manupiulate its distribution. We present integral forms for key quantities in the price of Asian option and its derivatives ( delta, gamma, theta, and vega ).

We propose a strategy to perform second-order enhancement using slope-limiters for the simultaneous linear advection of several scalar variables. Our strategy ensures a discrete min-max principle not only for each variable but also for any number of non-trivial combinations of them, which represent control variables. This problem arises in fluid mechanics codes using the Arbitrary Lagrange-Euler formalism, where the additional monotonicity property on control variables is required by physical considerations within the remap step.

Global existence of solutions is proved, in the case of positive cosmological constant and positive initial velocity of the cosmological expansion factor, on the three types of Friedman-Robertson-Walker space-times, and asymptotic behavior is investigated.

A nonlinear test model for filtering turbulent signals from partial observations of nonlinear slow-fast systems with multiple time scales is developed here. This model is a nonlinear stochastic real triad model with one slow mode. Despite the nonlinear and non-Gaussian features of the model, exact solution formulas are developed here for the mean and covariance. These formulas are utilized to develop a suite of statistically exact extended Kalman filters for the slow-fast system. Important practical issues such as filter performance with partial observations, which mix the slow and fast modes, model errors through linear filters for the fast modes, and the role of observation frequency and observational noise strength are assessed in unambiguous fashinon in the test model by utilizing these exact nonlinear statistics.

The aim of this work is to understand how urban traffic behavior, especially in cases of congestion, can be improved by an accurate choice of traffic coefficients. For this, we define three cost functionals, that measure average velocity, average travelling time and total flux of cars. The global optimal control problem for a complex network is hardly solvable both from analytical and numerical point of view. Thus, we focus on a simple junction with one incoming road and two outgoing roads (junctions of 1 \times 2 type), obtaining exact solutions to a siumple optimization problem. Then, we use such results at rach node of the network. The traffic evolution of some networks is then studied via simulations. In particular, it is shown that an appropriate choice of the traffic distributions can be useful in order to improve networks conditions.

We condider the aggregation equation with nonnegative initial data in L^1(R^n) \cap L^\infty(R^n) for n \ge 2. We assume that K is rotationally invariant, nonnegative, decaying at infinity, with at worst a Lipschitz point at the origin. We prove existence, uniqueness, and continuity of solutions. Finite time blow-up (in the L^\infty norm) of solutions is proved when the kernel has precisely a Lipschitz point at the origin.

The aim of this paper is to study a boundary value problem for a linear scalar equation with discontinuous coefficients. This kind of problem appears in the framework of the analysis of the linearized stability of a fluid flow with respect to small perturbations of the boundary data. The linear equation that we are interested in is obtained by linearizing the equations which govern the flow and it involves discontinuous coefficients and non trivial products. We present a direct approach based on the one introduced by Godlewski and Raviart which leads to measure solutions and gives a sense to these non trivial products and which yields simple numerical schemes that give good results.

We develop a method for 3D doubly periodic electromagnetic scattering. We adapt the Muller integral equation formulation of Maxwell's equations to the periodic problem, since it is a Fredholm equation of the second kind. We use Ewald splitting to efficiently calculate the periodic Green's functions. The approach is to regularize the singular Green's functions and to compute integrals with a trapezoid sum. Through asymptotic analysis near the singular point, we are able to idensity the largest part of the smoothing error and to subtract it out. The result is a method that is third order in the grid spacing size. We present results for various scatterers, including a test case for which exact solutions are known. The implemented method does indeed converge with third order accuracy. We present results for which the method successfully resolves Wood's anomaly resonances in transmission.

We prove the finite time blow-up for C^1 solutions of the attractive Euler-Poisson equations in R^n, n \ge 1, with and without background state, for a large set of 'generic' initial data. We characterize this super-critical set by tracing the spectral dynamics of the deformation and vorticity tensors.

We prove the existence of axially symmetric solutions to the Vlasov--Poisson system in a rotating setting for sufficiently small angular velocity. The constructed steady states depend on Jacobi's integral and the proof relies on an implicit function theorem for operators.

In this paper, we provide results about the large time behavior of integrodiferential equations appearing in the study of populations structured with respect to a quantitative (continuous) trait, which are submitted to selection (or competition).

Two types of filtering failure are the well known filter divergence where errors may exceed the size of the corresponding true chaotic attractor and the much more severe catastrophic filter divergence where solutions diverge to machine infinity in finite time. In this paper, we demonstrate that these failures occur in filtering the L-96 model, a nonlinear chaotic dissipative dynamical system with the absorbing ball property and quasi-Gaussian unimodal statistics. In particular, catastrophic filter divergence occurs in suitable parameter regimes for an ensemble Kalman filter when the noisy turbulent true solution signal is partially observed at sparse regular spatial locations.

With the above documentation, the main theme of this paper is to show that we can suppress the catastrophic filter divergence with a judicious model error strategy, that is, through a suitable linear stochastic model. This result confirms that the Gaussian assumption in the Kalman filter formulation, which is violated by most ensemble Kalman filters through the nonlinearity in the model, is a necessary condition to avoid catastrophic filter divergence. In a suitable range of chaotic regimes, adding model errors is not the best strategy when the true model is known. However, we find that there are several parameter regimes where the filtering performance in the presence of model errors with the stochastic model supersedes the performance in the perfect model simulation of the best ensemble Kalman filter considered here. Secondly, we also show that the advantage of the reduced Fourier domain filtering strategy is not simply through its numerical efficiency, but significant filtering accuracy is also gained through ignoring the correlation between the appropriate Fourier coefficients when the sparse observations are available in regular space locations.

We investigate the well-posedness of a coupled Stokes-Darcy model with Beavers-Joseph interface boundary conditions. In the steady-state case, the well-posedness is established under the assuumption of small coefficient in the Beavers-Joseph interface boundary condition. In the time-dependent case, the well-posedness is established via appropriate time discretization of the problem and a novel scaling of the system under isotropic media assumption. Such coupled systems are crucial to the study of subsurface flow problems, in particular, flows in karst aquifers.

A global-in-time existence theorem for classical solutions of the Vlasov-Darwin system is given under the assumption of smallness of the initial data. Furthermore it is shown that in case of spherical symmetry the system degenerates to the relativistic Vlasov-Poisson system.

The stability of solitary traveling waves in a general class of conservative nonlinear dispersive equations is discussed. A necessary condition for the exchange of stability of traveling waves is presented; an unstable eigenmode may bifurcate from the neutral translational mode only at relative extrema of the wave energy. This paper extends a result from Hamiltonian systems, and from a few integrable partial differential equations, to a broader class of conservative differential equations - with particular application to gravity-capillary surface waves.

In their classical 1937 paper, Kolmogorov, Petrovsky and Piskunov proved that for a particular class of reaction-diffusion equations on the line the solution of the initial value problem with the initial data in the form of a unit step propagates at long times with constant velocity equal to that of a certain special traveling wave solution. This type of propagation result has since been established for a number of general classes of reaction-diffusion-advection problems in cylinders. Here we show that actually in the problems without advection or in the presence of transverse advection by a potential flow these results do not rely on the specifics of the problem. Instead, they are a consequence of the fact that the considered equation is a gradient flow in an exponentially weighted L^2-space generated by a certain functional, when the dynamics is considered in the reference frame moving with constant velocity along the cylinder axis. We show that independently of the details of the problem only three propagation scenarios are possible in the above context: no propagation, a "pulled" front, or a "pushed" front. The choice of the scenario is completely characterized via a minimization problem.

The equation describing the non-stationary flow of an incompressible non-Newtonian fluid is approximated by the fully- and semi-implicit two-step backward differentiation formula (BDF). The stress tensor is assumed to be of $p$-structure such that the usual coercivity, growth, and monotonicity condition is fulfilled. Convergence of a piecewise polynomial prolongation of the discrete solution towards an exact weak solution is shown for the case $p \ge 1+ 2d/(d+2)$, where $d$ denotes the spatial dimension.

In this paper, we present a class of explicit numerical methods for stiff Ito stochastic differential equations (SDEs). These methods are as simple to program and to use as the well-known Euler-Maruyama method, but much more efficient for stiff SDEs. For such problems, it is well known that standard explicit methods face step-size reduction. While semi-implicit method can avoid these problems at the cost of solving (possibly large) nonlinear systems, we show that the step-size reduction phenomena can be reduced significantly for explicit methods by using stabilization techniques. Stabilized explicit numerical methods called S-ROCK (for stochastic orthogonal Runge-Kutta Chebyshev) have been introduced in [C.R. Acad. Sci. Paris, vol. 345, no. 10, 2007] as an alternative to (semi-)implicit methods for the solution of stiff stochastic systems. In this paper we discuss a genuine Ito version of the S-ROCK methods which avoid the use of transformation formulas from Stratonovish to Ito calculus. This is important for many applications as, for example, for the simulation of stiff chemical reactions or for dynamical systems used in finance. We present two families of methods for one-dimensional and multi-dimensional Wienner processes. We show that for stiff problems, significant improvement over classical explicit methods can be obtained. Convergence and stability properties of the methods are discussed and numerical examples and applications to the simulation of stiff chemical Langevin equations are presented.

We say that the vanishing viscosity limit holds in the classical sense if the velocity for a solution to the Navier-Stokes equations converges in the energy norm uniformly in time to the velocity for a solution to the Euler equations. We prove, for a bounded domain in dimension 2 or higher, that the vanishing viscosity limit holds in the classical sense if and only if a vortex sheet forms on the boundary.

We present an algorithm for interpolating the visible portions of a point cloud that are sampled from opaque objects in the environment. Our algorithm projects point clouds onto a sphere centered at the observing location and performs essentially non-oscillatory (ENO) interpolation of the project data. Curvatures of the clouding objects can be approximated and used in many ways. We demonstrate how our visibility formulation can be incorporated into novel algorithms for mapping unknown environments with a single or multiple observers, and target finding problems. A convergence proof is provided indicating suitability of our algorithm for some canonical types of environments. Various postprocessing optimization techniques are considered to obtain a more uniform exposure of the region along the path.