Metric based up-scaling

We consider divergence form elliptic operators in dimension $n\geq 2$ with $L^\infty$ coefficients. Although solutions of these operators are only H\"{o}lder continuous, we show that they are differentiable ($C^{1,\alpha}$) with respect to harmonic coordinates. It follows that numerical homogenization can be extended to situations where the medium has no ergodicity at small scales and is characterized by a continuum of scales by transferring a new metric in addition to traditional averaged (homogenized) quantities from subgrid scales into computational scales and error bounds can be given. This numerical homogenization method can also be used as a compression tool for differential operators.Comment: Final version. Accepted for publication in Communications on Pure and Applied Mathematics. Presented at CIMMS (March 2005), Socams 2005 (April), Oberwolfach, MPI Leipzig (May 2005), CIRM (July 2005). Higher resolution figures are available at http://www.acm.caltech.edu/~owhadi

Fractional Fourier approximations for potential gravity waves on deep water

In the framework of the canonical model of hydrodynamics, where fluid is assumed to be ideal and incompressible, waves are potential, two-dimensional, and symmetric, the authors have recently reported the existence of a new type of gravity waves on deep water besides well studied Stokes waves (Phys. Rev. Lett., 2002, v. 89, 164502). The distinctive feature of these waves is that horizontal water velocities in the wave crests exceed the speed of the crests themselves. Such waves were found to describe irregular flows with stagnation point inside the flow domain and discontinuous streamlines near the wave crests. Irregular flows produce a simple model for describing the initial stage of the formation of spilling breakers when a localized jet is formed at the crest following by generating whitecaps. In the present work, a new highly efficient method for computing steady potential gravity waves on deep water is proposed to examine the above results in more detail. The method is based on the truncated fractional approximations for the velocity potential in terms of the basis functions $1/\bigr(1-\exp(y_0-y-ix)\bigl)^n$, $y_0$ being a free parameter. The non-linear transformation of the horizontal scale $x = \chi - \gamma \sin\chi, 0<\gamma<1,$ is additionally applied to concentrate a numerical emphasis on the crest region of a wave for accelerating the convergence of the series. Fractional approximations were employed for calculating both steep Stokes waves and irregular flows. For lesser computational time, the advantage in accuracy over ordinary Fourier expansions in terms the basis functions $\exp\bigl(n (y+ix)\bigr)$ was found to be from one to ten decimal orders depending on the wave steepness and flow parameters.Comment: 14 pages, 8 figures, submitted to Nonlinear Processes in Geophysic

On the Szeg\"o-Asymptotics for Doubly-Dispersive Gaussian Channels

We consider the time-continuous doubly-dispersive channel with additive Gaussian noise and establish a capacity formula for the case where the channel correlation operator is represented by a symbol which is periodic in time and fulfills some further integrability and smoothness conditions. The key to this result is a new Szeg\"o formula for certain pseudo-differential operators. The formula justifies the water-filling principle along time and frequency in terms of the time--continuous time-varying transfer function (the symbol).Comment: 5 pages, to be presented at ISIT 2011, minor typos corrected, references update

A strategy to suppress recurrence in grid-based Vlasov solvers

In this paper we propose a strategy to suppress the recurrence effect present in grid-based Vlasov solvers. This method is formulated by introducing a cutoff frequency in Fourier space. Since this cutoff only has to be performed after a number of time steps, the scheme can be implemented efficiently and can relatively easily be incorporated into existing Vlasov solvers. Furthermore, the scheme proposed retains the advantage of grid-based methods in that high accuracy can be achieved. This is due to the fact that in contrast to the scheme proposed by Abbasi et al. no statistical noise is introduced into the simulation. We will illustrate the utility of the method proposed by performing a number of numerical simulations, including the plasma echo phenomenon, using a discontinuous Galerkin approximation in space and a Strang splitting based time integration

Non-oscillatory spectral Fourier methods for shock wave calculations

A non-oscillatory spectral Fourier method is presented for the solution of hyperbolic partial differential equations. The method is based on adding a nonsmooth function to the trigonometric polynomials which are the usual basis functions for the Fourier method. The high accuracy away from the shock is enhanced by using filters. Numerical results confirm that no oscillations develop in the solution. Also, the accuracy of the spectral solution of the inviscid Burgers equation is shown to be higher than a fixed order