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We study the limiting dynamics of the Euler Alignment system with a smooth, heavy-tailed interaction kernel $\phi$ and unidirectional velocity $\mathbf{u} = (u, 0, \ldots, 0)$. We demonstrate a striking correspondence between the entropy function $e_0 = \partial_1 u_0 + \phi*\rho_0$ and the limiting 'concentration set', i.e., the support of the singular part of the limiting density measure. In a typical scenario, a flock experiences aggregation toward a union of $C^1$ hypersurfaces: the image of the zero set of $e_0$ under the limiting flow map. This correspondence also allows us to make statements about the fine properties associated to the limiting dynamics, including a sharp upper bound on the dimension of the concentration set, depending only on the smoothness of $e_0$. In order to facilitate and contextualize our analysis of the limiting density measure, we also include an expository discussion of the wellposedness, flocking, and stability of the Euler Alignment system, most of which is new., Comment: 20 pages, 1 figure

We continue our study of hydrodynamic models of self-organized evolution of agents with singular interaction kernel ϕ ( x ) = | x | − ( 1 + α ) . Following our works Shvydkoy and Tadmor (2017) [1] , [2] which focused on the range 1 ≤ α 2 , and Do et al. (2017) which covered the range 0 α 1 , in this paper we revisit the latter case and give a short(-er) proof of global in time existence of smooth solutions, together with a full description of their long time dynamics. Specifically, we prove that starting from any initial condition in ( ρ 0 , u 0 ) ∈ H 2 + α × H 3 , the solution approaches exponentially fast to a flocking state solution consisting of a wave ρ = ρ ∞ ( x − t u ) traveling with a constant velocity determined by the conserved average velocity u . The convergence is accompanied by exponential decay of all higher order derivatives of u .

We consider the shallow water equations with non-flat bottom topography. The smooth solutions of these equations are energy conservative, whereas weak solutions are energy stable. The equations possess interesting steady states of lake at rest as well as moving equilibrium states. We design energy conservative finite volume schemes which preserve (i) the lake at rest steady state in both one and two space dimensions, and (ii) one-dimensional moving equilibrium states. Suitable energy stable numerical diffusion operators, based on energy and equilibrium variables, are designed to preserve these two types of steady states. Several numerical experiments illustrating the robustness of the energy preserving and energy stable well-balanced schemes are presented.

We improve the recent result of Chae and Tadmor (2008) [10] proving a one-sided threshold condition which leads to a finite-time breakdown of the Euler–Poisson equations in arbitrary dimension n .

The computations reported in this paper demonstrate the remarkable versatility of central schemes as black-box, Jacobian-free solvers for ideal magnetohydrodynamics (MHD) equations. Here we utilize a family of high-resolution, non-oscillatory central schemes for the approximate solution of the one- and two-dimensional MHD equations. We present simulations based on staggered grids of several MHD prototype problems. Solution of one-dimensional shock-tube problems is carried out using second- and third-order central schemes [Numer. Math. 79 (1998) 397; J. Comput. Phys. 87 (2) (1990) 408], and we use the second-order central scheme [SIAM J. Sci Comput. 19 (6) (1998) 1892] which is adapted for the solution of the two-dimensional Kelvin-Helmholtz and Orszag-Tang problems. A qualitative comparison reveals an excellent agreement with previous results based on upwind schemes. Central schemes, however, require little knowledge about the eigenstructure of the problem in fact, we even avoid an explicit evaluation of the corresponding Jacobians, while at the same time they eliminate the need for dimensional splitting.

In this paper we construct, analyze and implement a new procedure for the spectral approximations of nonlinear conservation laws. It is well known that using spectral methods for nonlinear conservation laws will result in the formation of the Gibbs phenomenon once spontaneous shock discontinuities appear in the solution. These spurious oscillations will in turn lead to loss of resolution and render the standard spectral approximations unstable. The Spectral Viscosity (SV-) method (Tadmor, 1989) was developed to stabilize the spectral method by adding a spectrally small amount of high-frequencies diffusion carried out in the dual space. The resulting SV-approximation is stable without sacrificing spectral accuracy. The SV-method recovers a spectrally accurate approximation to the projection of the entropy solution; the exact projection, however, is at best a first order approximation to the exact solution as a result of the formation of the shock discontinuities. The issue of spectral resolution is addressed by post-processing the SV-solution to remove the spurious oscillations at the discontinuities, as well as increase the first-order—O.1=N/ accuracy away from the shock discontinuities. Successful post-processing methods have been developed to eliminate the Gibbs phenomenon and recover spectral accuracy for the SV-approximation. However, such reconstruction methods require a priori knowledge of the locations of the shock discontinuities. Therefore, the detection of these discontinuities is essential to obtain an overall spectrally accurate solution. To this end, we employ the recently constructed enhanced edge detectorsbased on appropriate concentration factors (Gelb and Tadmor, 1999). Once the edges of these discontinuities are identified, we can utilize a post-processing reconstruction method, and show that the post-processed SV-solution recovers the exact entropy solution with remarkably high-resolution. We apply our new numerical method, the Enhanced SV-method, to two numerical examples, the scalar periodic Burgers’ equation and the one-dimensional system of Euler equations of gas dynamics. Both approximations exhibit high accuracy and resolution to the exact entropy solution. © 2000 IMACS. Published by Elsevier Science B.V. All rights reserved.

We introduce a new high-resolution central scheme for multidimensional Hamilton?Jacobi equations. The scheme retains the simplicity of the non-oscillatory central schemes developed by C.-T. Lin and E. Tadmor (in press, SIAM J. Sci. Comput.), yet it enjoys a smaller amount of numerical viscosity, independent of 1/?t. By letting ?t?0 we obtain a new second-order central scheme in the particularly simple semi-discrete form, along the lines of the new semi-discrete central schemes recently introduced by the authors in the context of hyperbolic conservation laws. Fully discrete versions are obtained with appropriate Runge?Kutta solvers. The smaller amount of dissipation enables efficient integration of convection-diffusion equations, where the accumulated error is independent of a small time step dictated by the CFL limitation. The scheme is non-oscillatory thanks to the use of nonlinear limiters. Here we advocate the use of such limiters on second discrete derivatives, which is shown to yield an improved high resolution when compared to the usual limitation of first derivatives. Numerical experiments demonstrate the remarkable resolution obtained by the proposed new central scheme.

Central schemes may serve as universal finite-difference methods for solving nonlinear convection?diffusion equations in the sense that they are not tied to the specific eigenstructure of the problem, and hence can be implemented in a straightforward manner as black-box solvers for general conservation laws and related equations governing the spontaneous evolution of large gradient phenomena. The first-order Lax?Friedrichs scheme (P. D. Lax, 1954) is the forerunner for such central schemes. The central Nessyahu?Tadmor (NT) scheme (H. Nessyahu and E. Tadmor, 1990) offers higher resolution while retaining the simplicity of the Riemann-solver-free approach. The numerical viscosity present in these central schemes is of order O((?x)2r/?t). In the convective regime where ?t~?x, the improved resolution of the NT scheme and its generalizations is achieved by lowering the amount of numerical viscosity with increasing r. At the same time, this family of central schemes suffers from excessive numerical viscosity when a sufficiently small time step is enforced, e.g., due to the presence of degenerate diffusion terms.In this paper we introduce a new family of central schemes which retain the simplicity of being independent of the eigenstructure of the problem, yet which enjoy a much smaller numerical viscosity (of the corresponding order O(?x)2r?1)). In particular, our new central schemes maintain their high-resolution independent of O(1/?t), and letting ?t ? 0, they admit a particularly simple semi-discrete formulation. The main idea behind the construction of these central schemes is the use of more precise information of the local propagation speeds. Beyond these CFL related speeds, no characteristic information is required. As a second ingredient in their construction, these central schemes realize the (nonsmooth part of the) approximate solution in terms of its cell averages integrated over the Riemann fans of varying size.The semi-discrete central scheme is then extended to multidimensional problems, with or without degenerate diffusive terms. Fully discrete versions are obtained with Runge?Kutta solvers. We prove that a scalar version of our high-resolution central scheme is nonoscillatory in the sense of satisfying the total-variation diminishing property in the one-dimensional case and the maximum principle in two-space dimensions. We conclude with a series of numerical examples, considering convex and nonconvex problems with and without degenerate diffusion, and scalar and systems of equations in one- and two-space dimensions. Time evolution is carried out by the third- and fourth-order explicit embedded integration Runge?Kutta methods recently proposed by A. Medovikov (1998). These numerical studies demonstrate the remarkable resolution of our new family of central scheme.

Many of the recently developed high resolution schemes for hyperbolic conservation laws are based on upwind differencing. The building block for these schemes is the averaging of an appropriate Godunov solver; its time consuming part involves the field-by-field decomposition which is required in order to identify the direction of the wind. Instead, the use of the more robust Lax-Friedrichs (LxF) solver is proposed. The main advantage is simplicity: no Riemann problems are solved and hence field-by-field decompositions are avoided. The main disadvantage is the excessive numerical viscosity typical to the LxF solver. This is compensated for by using high-resolution MUSCL-type interpolants. Numerical experiments show that the quality of results obtained by such convenient central differencing is comparable with those of the upwind schemes.

Let u(@?x, t) be a weak solution of the Euler equation, governing the inviscid polytropic gas dynamics; in addition, u(@?x, t) is assumed to respect the usual entropy conditions connected with the conservative Euler equations. We show that such entropy solutions of the gas dynamics equations satisfy a minimum entropy principle, namely, that the spatial minimum of their specific entropy, Ess inf"@?"xS(u(@?x, t)), is an increasing function of time. This principle equally applies to discrete approximations of the Euler equations such as the Godunov-type and Lax-Friedrichs schemes. Our derivation of this minimum principle makes use of the fact that there is a family of generalized entropy functions connected with the conservative Euler equations.

Using a simple symmetrizability criterion, we show that symmetric systems of conservation laws are equipped with a one-parameter family of entropy functions.

We study some properties of the numerical radius of matrices with non-negative entries, and explicit ways to compute it. We also characterize positive matrices with equal spectral and numerical radii, i.e., positive spectral matrices.

We show that any spectrally dominant vector norm on the vector space of matrices which is invariant under isometries dominates the numerical radius r (·). Thus, the celebrated Lax-Wendroff stability condition, r (·)⩽1, defines a maximal isometrically invariant stable set.

We give a brief account of the numerical radius of a linear bounded operator on a Hilbert space and some of its better-known properties. Both finite- and infinite- dimensional aspects are discussed, as well as applications to stability theory of finite-difference approximations for hyperbolic initial-value problems.