### Browsing by Author "TADMOR, EITAN"

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Item CENTRAL DISCONTINUOUS GALERKIN METHODS ON OVERLAPPING CELLS WITH A NONOSCILLATORY HIERARCHICAL RECONSTRUCTION(Copyright: Society for Industrial and Applied Mathematics, 2007) LIU, YINGJIE; SHU, CHI-WANG; TADMOR, EITAN; ZHANG, MENGPINGThe 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.Item CRITICAL THRESHOLDS IN 2D RESTRICTED EULER–POISSON EQUATIONS(Copyright: Society for Industrial and Applied Mathematics, 2003) LIU, HAILIANG; TADMOR, EITANWe provide a complete description of the critical threshold phenomenon for the twodimensional localized Euler–Poisson equations, introduced by the authors in [Comm. Math. Phys., 228 (2002), pp. 435–466]. Here, the questions of global regularity vs. finite-time breakdown for the two-dimensional (2D) restricted Euler–Poisson solutions are classified in terms of precise explicit formulae, describing a remarkable variety of critical threshold surfaces of initial configurations. In particular, it is shown that the 2D critical thresholds depend on the relative sizes of three quantities: the initial density, the initial divergence, and the initial spectral gap, that is, the difference between the two eigenvalues of the 2 × 2 initial velocity gradient.Item CRITICAL THRESHOLDS IN A CONVOLUTION MODEL FOR NONLINEAR CONSERVATION LAWS(Copyright: Society for Industrial and Applied Mathematics, 2001) LIU, HAILIANG; TADMOR, EITANIn this work we consider a convolution model for nonlinear conservation laws.Due to the delicate balance between the nonlinear convection and the nonlocal forcing, this model allows for narrower shock layers than those in the viscous Burgers’ equation and yet exhibits the conditional finite time breakdown as in the damped Burgers’ equation.W e show the critical threshold phenomenon by presenting a lower threshold for the breakdown of the solutions and an upper threshold for the global existence of the smooth solution.The threshold condition depends only on the relative size of the minimum slope of the initial velocity and its maximal variation.W e show the exact blow-up rate when the slope of the initial profile is below the lower threshold.W e further prove the L1 stability of the smooth shock profile, provided the slope of the initial profile is above the critical threshold.Item Detection of Edges in Spectral Data II. Nonlinear Enhancement(Copyright: Society for Industrial and Applied Mathematics, 2000) GELB, ANNE; TADMOR, EITANWe discuss a general framework for recovering edges in piecewise smooth functions with finitely many jump discontinuities, where [f](x) := f(x+)−f(x−) ≠ 0. Our approach is based on two main aspects—localization using appropriate concentration kernels and separation of scales by nonlinear enhancement. To detect such edges, one employs concentration kernels, K_𝛆(·), depending on the small scale 𝛆. Itis shown that odd kernels, properly scaled, and admissible (in the sense of having small W−1,∞- moments of order O(𝛆)) satisfy K_𝛆 ∗ f(x) = [f](x) + O(𝛆), thus recovering both the location and amplitudes of all edges. As an example we consider general concentration kernels of the form KσN (t) = 𝝨σ(k/N) sin kt to detect edges from the first 1/𝛆 = N spectral modes of piecewise smooth f’s. Here we improve in generality and simplicity over our previous study in [A. Gelb and E. Tadmor, Appl. Comput. Harmon. Anal., 7 (1999), pp. 101–135]. Both periodic and nonperiodic spectral projections are considered. We identify, in particular, a new family of exponential factors, σexp(·), with superior localization properties. The other aspect of our edge detection involves a nonlinear enhancement procedure which is based on separation of scales between the edges, where K_𝛆 ∗ f(x) ∼ [f](x) ≠ 0, and the smooth regions where K_𝛆 ∗ f = O(𝛆) ∼ 0. Numerical examples demonstrate that by coupling concentration kernels with nonlinear enhancement one arrives at effective edge detectors.Item LONG-TIME EXISTENCE OF SMOOTH SOLUTIONS FOR THLong time existence of smooth solutions for the rapidly rotating shallow-water and Euler equationsE(Copyright: Society for Industrial and Applied Mathematics, 2008) CHENG, BIN; TADMOR, EITANWe study the stabilizing effect of rotational forcing in the nonlinear setting of two-dimensional 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 observations.Item A MULTISCALE IMAGE REPRESENTATION USING HIERARCHICAL (BV,L2) DECOMPOSITIONS(Copyright: Society for Industrial and Applied Mathematics, 2004) TADMOR, EITAN; NEZZAR, SUZANNE; VESE, LUMINITAWe propose a new multiscale image decomposition which offers a hierarchical, adaptive representation for the different features in general images. The starting point is a variational decomposition of an image, f = u0 + v0, where [u0, v0] is the minimizer of a J-functional, J(f, λ0; X, Y ) = infu+v=f u X + λ0 v p Y . Such minimizers are standard tools for image manipulations (e.g., denoising, deblurring, compression); see, for example, [M. Mumford and J. Shah, Proceedings of the IEEE Computer Vision Pattern Recognition Conference, San Francisco, CA, 1985] and [L. Rudin, S. Osher, and E. Fatemi, Phys. D, 60 (1992), pp. 259–268]. Here, u0 should capture “essential features” of f which are to be separated from the spurious components absorbed by v0, and λ0 is a fixed threshold which dictates separation of scales. To proceed, we iterate the refinement step [uj+1, vj+1] = arginf J(vj, λ02j ), leading to the hierarchical decomposition, f = k j=0 uj + vk. We focus our attention on the particular case of (X, Y) = (BV,L2) decomposition. The resulting hierarchical decomposition, f ∼ j uj , is essentially nonlinear. The questions of convergence, energy decomposition, localization, and adaptivity are discussed. The decomposition is constructed by numerical solution of successive Euler–Lagrange equations. Numerical results illustrate applications of the new decomposition to synthetic and real images. Both greyscale and color images are considered.Item Pointwise error estimates for relaxation approximations to conservation laws(Copyright: Society for Industrial and Applied Mathematics, 2000) TADMOR, EITAN; TANG, TAOWe obtain sharp pointwise error estimates for relaxation approximation to scalar conservation laws with piecewise smooth solutions. We first prove that the first-order partial derivatives for the perturbation solutions are uniformly upper bounded (the so-called Lip+ stability). A one-sided interpolation inequality between classical L1 error estimates and Lip+ stability bounds enables us to convert a global L1 result into a (nonoptimal) local estimate. Optimal error bounds on the weighted error then follow from the maximum principle for weakly coupled hyperbolic systems. The main difficulties in obtaining the Lip+ stability and the optimal pointwise errors are how to construct appropriate “difference functions” so that the maximum principle can be applied.Item RECOVERY OF EDGES FROM SPECTRAL DATA WITH NOISE—A NEW PERSPECTIVE(Copyright: Society for Industrial and Applied Mathematics, 2008) ENGELBERG, SHLOMO; TADMOR, EITANWe 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.Item SPECTRAL VANISHING VISCOSITY METHOD FOR NONLINEAR CONSERVATION LAWS(Copyright: Society for Industrial and Applied Mathematics, 2001) GUO, BEN-YU; MA, HE-PING; TADMOR, EITANWe propose a new spectral viscosity(SV) scheme for the accurate solution of nonlinear conservation laws. It is proved that the SV solution converges to the unique entropysolution under appropriate reasonable conditions. The proposed SV scheme is implemented directlyon high modes of the computed solution. This should be compared with the original nonperiodic SV scheme introduced byMada y, Ould Kaber, and Tadmor in [SIAM J. Numer. Anal., 30 (1993), 321–342], where SV is activated on the derivative of the SV solution. The new proposed SV method could be viewed as a correction of the former, and it offers an improvement which is confirmed byour numerical experiments. A postprocessing method is implemented to greatlyenhance the accuracyof the computed SV solution. The numerical results show the efficiencyof the new method.