Perfect derivatives, conservative differences and entropy stable computation of hyperbolic conservation laws
Eitan Tadmor
Discrete & Continuous Dynamical Systems - A 2016, 36(8): 4579-4598 doi: 10.3934/dcds.2016.36.4579
Entropy stability plays an important role in the dynamics of nonlinear systems of hyperbolic conservation laws and related convection-diffusion equations. Here we are concerned with the corresponding question of numerical entropy stability --- we review a general framework for designing entropy stable approximations of such systems. The framework, developed in [28,29] and in an ongoing series of works [30,6,7], is based on comparing numerical viscosities to certain entropy-conservative schemes. It yields precise characterizations of entropy stability which is enforced in rarefactions while keeping sharp resolution of shocks.
    We demonstrate this approach with a host of second-- and higher--order accurate schemes, ranging from scalar examples to the systems of shallow-water, Euler and Navier-Stokes equations. We present a family of energy conservative schemes for the shallow-water equations with a well-balanced description of their steady-states. Numerical experiments provide a remarkable evidence for the different roles of viscosity and heat conduction in forming sharp monotone profiles in Euler equations, and we conclude with the computation of entropic measure-valued solutions based on the class of so-called TeCNO schemes --- arbitrarily high-order accurate, non-oscillatory and entropy stable schemes for systems of conservation laws.
keywords: Entropy conservative schemes measure-valued solutions. Euler and Navier-Stokes equations shallow water equations entropy stability energy preserving schemes numerical viscosity
Eulerian dynamics with a commutator forcing Ⅱ: Flocking
Roman Shvydkoy Eitan Tadmor
Discrete & Continuous Dynamical Systems - A 2017, 37(11): 5503-5520 doi: 10.3934/dcds.2017239

We continue our study of one-dimensional class of Euler equations, introduced in [11], driven by a forcing with a commutator structure of the form $[{\mathcal L}_φ, u](ρ)$, where $u$ is the velocity field and ${\mathcal L}_φ$ belongs to a rather general class of convolution operators depending on interaction kernels $φ$.

In this paper we quantify the large-time behavior of such systems in terms of fast flocking, for two prototypical sub-classes of kernels: bounded positive $φ$'s, and singular $φ(r) = r^{-(1+α)}$ of order $α∈ [1, 2)$ associated with the action of the fractional Laplacian ${\mathcal L}_φ=-(-\partial_{xx})^{α/2}$. Specifically, we prove fast velocity alignment as the velocity $u(·, t)$ approaches a constant state, $u \to \bar{u}$, with exponentially decaying slope and curvature bounds $|{u_x}( \cdot ,t){|_\infty } + |{u_{xx}}( \cdot ,t){|_\infty }\lesssim{e^{ - \delta t}}$. The alignment is accompanied by exponentially fast flocking of the density towards a fixed traveling state $ρ(·, t) -{ρ_{∞}}(x -\bar{u} t) \to 0$.

keywords: Flocking alignment fractional dissipation Cucker-Smale Motsch-Tadmor critical thresholds
From particle to kinetic and hydrodynamic descriptions of flocking
Seung-Yeal Ha Eitan Tadmor
Kinetic & Related Models 2008, 1(3): 415-435 doi: 10.3934/krm.2008.1.415
We discuss the Cucker-Smale's (C-S) particle model for flocking, deriving precise conditions for flocking to occur when pairwise interactions are sufficiently strong long range. We then derive a Vlasov-type kinetic model for the C-S particle model and prove it exhibits time-asymptotic flocking behavior for arbitrary compactly supported initial data. Finally, we introduce a hydrodynamic description of flocking based on the C-S Vlasov-type kinetic model and prove flocking behavior without closure of higher moments.
keywords: hydrodynamic formulation. moments flocking particles kinetic formulation
Multiscale image representation using novel integro-differential equations
Eitan Tadmor Prashant Athavale
Inverse Problems & Imaging 2009, 3(4): 693-710 doi: 10.3934/ipi.2009.3.693
Motivated by the hierarchical multiscale image representation of Tadmor et. al., [25], we propose a novel integro-differential equation (IDE) for a multiscale image representation. To this end, one integrates in inverse scale space a succession of refined, recursive 'slices' of the image, which are balanced by a typical curvature term at the finer scale. Although the original motivation came from a variational approach, the resulting IDE can be extended using standard techniques from PDE-based image processing. We use filtering, edge preserving and tangential smoothing to yield a family of modified IDE models with applications to image denoising and image deblurring problems. The IDE models depend on a user scaling function which is shown to dictate the BV properties of the residual error. Numerical experiments demonstrate application of the IDE approach to denoising and deblurring.
keywords: natural images multiscale representation variational problem inverse scale deblurring integro-differential equation total variation energy decomposition. denoising

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