DEPARTMENT OF MATHEMATICS AND STATISTICS

23/10/13 George Hagstrom, Courant, Phase Space Pattern Formation and Stability of Inhomogeneous Equilibria of Hamiltonian Continuous Media Field Theories .

There are a wide variety of 1+1 Hamiltonian continuous media field theories that exhibit phase space pattern formation. In plasma physics, the most famous of these is the Vlasov-Poisson equation, but other examples include the incompressible Euler and barotropic equations in two-dimensions and the Hamiltonian Mean Field model. One of the characteristic phenomenon that occurs in systems described by these equations is the formation of cat's eye patterns in phase space as a result of the nonlinear saturation of instabilities. Corresponding to each cat's eye is a spatially inhomogeneous equilibrium solution of the underlying model, in plasma physics these are called BGK modes, but analogous solutions exist in all of the above systems. We analyze the stability of inhomogeneous equilibria in the Hamiltonian Mean Field model and in the Single Wave model, which is an equation that was derived to provide a model of the formation of electron holes in plasmas. We use action angle variables and the properties of elliptic functions to analyze the resulting dispersion relation and construct stability criterion for inhomogeneous equilibria and study the behavior of solutions near these equilibria.

30/10/13 Catalin Turc, Mathematics, NJIT, Well conditioned boundary integral equations for solution of frequency domain electromgnetic scattering problems at high frequencies.

We will discuss in this talk a general framework capable of producing integral equation formulations for freqency domain electromagnetic scattering problems that require small numbers of Krylov subspace iterations for high frequencies. These integral formulations are amenable to both Galerkin and Nystrom discretizations, and consistenly outperform the classical formulations. In addition, these formulations enjoy certain surprising coercivity properties that allow for an error analysis of the Galerkin and Nystrom discretizations that exhibits explicit frequency dependent bounds. (with Y. Boubendir)

06/11/13 Matt Causely, Mathematics, Michigan State, Higher-order A-stable schemes for the wave equation through successive convolution .

We present a new methodology for solving the wave equation, using a boundary integral solver which employs fast convolution, to achieve an O(N) algorithm. The method can address all standard boundary conditions, including out ow, and also works on complex boundaries, which are embedded into a Cartesian mesh. The purpose of this solver is to solve multi-scale plasma problems in complex geometries, where time steps in excess of the Courant-Friedrichs-Lewy (CFL) stability condition must be taken. To this extent, the solutions we obtain are A-stable, achieve higher order accuracy in space, and second order accuracy in time. Recently, a means to extend the accuracy to orders higher than 2, without sacrificing A-stability, was discovered. This result is obtained by means of successive convolution, and is a systematic approach to incorporating higher order error terms in the numerical scheme. The stability properties follow from the fact that higher derivatives are approximated with iterated applications of a convolution algorithm, as opposed to numerically unstable finite difference approximations. The accuracy, stability and efficiency of the solver are addressed with several numerical examples. Due to the efficiency and exibility of the solver, domain decomposition and parallelization of the algorithm is the subject of future work.

13/11/13 Bogdan Nita, Mathematics, Montclair State, Seismic imaging and inversion using scattering theory .

Inverse scattering has recently proved to be an effective tool for processing seismic data with the goal of determining the structure (imaging) and the properties (inversion) of a medium under investigation. The method takes the form of a series, the inverse scattering series, which, when convergent, will output this information directly from the full data set recorded on a measurement surface without any assumed knowledge about the medium. In this talk I describe an algorithm, derived as a subseries of the full inverse scattering series, for geophysical imaging and amplitude correction. The algorithm is shown to converge to a closed form, independently of the parameters involved in the problem. Analytic and numerical one dimensional examples show excellent results in finding both the location of interfaces and the amplitude of acoustic reflections including in the cases in which data is noisy or it's missing low frequency components.

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