DRUM Collection: Mathematics Research Works

SPECTRAL METHODS FOR HYPERBOLIC PROBLEMS

2008-11-15T03:30:34Z

Title: SPECTRAL METHODS FOR HYPERBOLIC PROBLEMS Authors: Tadmor, Eitan Abstract: We review several topics concerning spectral approximations of time-dependent problems, primarily | the accuracy and stability of Fourier and Chebyshev methods for the approximate solutions of hyperbolic systems. To make these notes self contained, we begin with a very brief overview of Cauchy problems. Thus, the main focus of the �rst part is on hyperbolic systems which are dealt with two (related) tools: the energy method and Fourier analysis. The second part deals with spectral approximations. Here we introduce the main ingredients of spectral accuracy, Fourier and Chebyshev interpolants, aliasing, di�erentiation matrices ... The third part is devoted to Fourier method for the approximate solution of periodic systems. The questions of stability and convergence are answered by combining ideas from the �rst two sections. In this context we highlight the role of aliasing and smoothing; in particular, we explain how the lack of resolution might excite small scales weak instability, which is avoided by high modes smoothing. The forth and �nal part deals with non-periodic problems. We study the stability of the Chebyshev method, paying special attention to the intricate issue of the CFL stability restriction on the permitted time-step.

RECOVERY OF EDGES FROM SPECTRAL DATA WITH NOISE—A NEW PERSPECTIVE

2008-11-07T03:30:57Z

Title: RECOVERY OF EDGES FROM SPECTRAL DATA WITH NOISE—A NEW PERSPECTIVE Authors: ENGELBERG, SHLOMO; TADMOR, EITAN Abstract: We consider the problem of detecting edges—jump discontinuities in piecewise smooth functions from their N-degree spectral content, which is assumed to be corrupted by noise. There are three scales involved: the “smoothness” scale of order 1/N, the noise scale of order √η, and the O(1) scale of the jump discontinuities. We use concentration factors which are adjusted to the standard deviation of the noise √η � 1/N in order to detect the underlying O(1)-edges, which are separated from the noise scale √η � 1.

LONG-TIME EXISTENCE OF SMOOTH SOLUTIONS FOR THLong time existence of smooth solutions for the rapidly rotating shallow-water and Euler equationsE

2008-11-04T03:31:05Z

Title: LONG-TIME EXISTENCE OF SMOOTH SOLUTIONS FOR THLong time existence of smooth solutions for the rapidly rotating shallow-water and Euler equationsE Authors: CHENG, BIN; TADMOR, EITAN Abstract: We study the stabilizing effect of rotational forcing in the nonlinear setting of twodimensional shallow-water and more general models of compressible Euler equations. In [Phys. D, 188 (2004), pp. 262–276] Liu and Tadmor have shown that the pressureless version of these equations admit a global smooth solution for a large set of subcritical initial configurations. In the present work we prove that when rotational force dominates the pressure, it prolongs the lifespan of smooth solutions for t <∼ ln(δ−1); here δ � 1 is the ratio of the pressure gradient measured by the inverse squared Froude number, relative to the dominant rotational forces measured by the inverse Rossby number. Our study reveals a “nearby” periodic-in-time approximate solution in the small δ regime, upon which hinges the long-time existence of the exact smooth solution. These results are in agreement with the close-to-periodic dynamics observed in the “near-inertial oscillation” (NIO) regime which follows oceanic storms. Indeed, our results indicate the existence of a smooth, “approximate periodic” solution for a time period of days, which is the relevant time period found in NIO obesrvations.

CENTRAL DISCONTINUOUS GALERKIN METHODS ON OVERLAPPING CELLS WITH A NONOSCILLATORY HIERARCHICAL RECONSTRUCTION

2008-11-04T03:31:04Z

Title: CENTRAL DISCONTINUOUS GALERKIN METHODS ON OVERLAPPING CELLS WITH A NONOSCILLATORY HIERARCHICAL RECONSTRUCTION Authors: LIU, YINGJIE; SHU, CHI-WANG; TADMOR, EITAN; ZHANG, MENGPING Abstract: The central scheme of Nessyahu and Tadmor [J. Comput. Phys., 87 (1990), pp. 408–463] solves hyperbolic conservation laws on a staggered mesh and avoids solving Riemann problems across cell boundaries. To overcome the difficulty of excessive numerical dissipation for small time steps, the recent work of Kurganov and Tadmor [J. Comput. Phys., 160 (2000), pp. 241–282] employs a variable control volume, which in turn yields a semidiscrete nonstaggered central scheme. Another approach, which we advocate here, is to view the staggered meshes as a collection of overlapping cells and to realize the computed solution by its overlapping cell averages. This leads to a simple technique to avoid the excessive numerical dissipation for small time steps [Y. Liu, J. Comput. Phys., 209 (2005), pp. 82–104]. At the heart of the proposed approach is the evolution of two pieces of information per cell, instead of one cell average which characterizes all central and upwind Godunov-type finite volume schemes. Overlapping cells lend themselves to the development of a central-type discontinuous Galerkin (DG) method, following the series of works by Cockburn and Shu [J. Comput. Phys., 141 (1998), pp. 199–224] and the references therein. In this paper we develop a central DG technique for hyperbolic conservation laws, where we take advantage of the redundant representation of the solution on overlapping cells. The use of redundant overlapping cells opens new possibilities beyond those of Godunov-type schemes. In particular, the central DG is coupled with a novel reconstruction procedure which removes spurious oscillations in the presence of shocks. This reconstruction is motivated by the moments limiter of Biswas, Devine, and Flaherty [Appl. Numer. Math., 14 (1994), pp. 255–283] but is otherwise different in its hierarchical approach. The new hierarchical reconstruction involves a MUSCL or a second order ENO reconstruction in each stage of a multilayer reconstruction process without characteristic decomposition. It is compact, easy to implement over arbitrary meshes, and retains the overall preprocessed order of accuracy while effectively removing spurious oscillations around shocks.