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

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 a positive cosmological constant and positive initial velocity of the cosmological expansion factor on the three types of Friedman- Robertson-Walker space-time, 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, two fast modes, and catalytic nonlinear interaction of the fast modes depending on the 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 fashion in the test model by utilizing these exact nonlinear statistics.

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 involves discontinuous coefficients and nontrivial products. We present a direct approach based on the one introduced by Godlewski and Raviart, which leads to measure solutions, gives a sense of these nontrivial products, and yields simple numerical schemes that give good results.

We develop a method for 3D doubly periodic electromagnetic scattering. We adapt the Müller 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 trapezoidal sum. Through asymptotic analysis near the singular point, we are able to identify 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 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 long time behavior of integrodifferential equations appearing in the study of populations structured with respect to a quantitative (continuous) trait, which are submitted to selection (or competition).

A global-in-time existence theorem for classical solutions of the Vlasov-Darwin sys- tem 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 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 difficult to solve both from analytical and numerical points of view. Thus, we focus on a simple junction with one incoming road and two outgoing roads (junctions of 1×2 type), obtaining exact solutions to a simple optimization problem. Then, we use such results at each 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 network conditions.

We prove the finite time blow-up for $C^1$ solutions of the attractive Euler-Poisson equations in $\Bbb R^{2}$, $n\geq1$, with and without background state, for a large set of ’generic’ initial data. We characterize this supercritical set by tracing the spectral dynamics of the deformation and vorticity tensors.

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.