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We study the long-time hydrodynamic behavior of systems of multi-species which arise from agent-based description of alignment dynamics. The interaction between species is governed by an array of symmetric communication kernels. We prove that the crowd of different species flocks towards the mean velocity if (i) cross interactions form a heavy-tailed connected array of kernels, while (ii) self-interactions are governed by kernels with singular heads. The main new aspect here is that flocking behavior holds without closure assumption on the specific form of pressure tensors. Specifically, we prove the long-time flocking behavior for connected arrays of multi-species, with self-interactions governed by entropic pressure laws (see E. Tadmor [Bull. Amer. Math. Soc. (2023), to appear]) and driven by fractional p p -alignment. In particular, it follows that such multi-species hydrodynamics approaches a mono-kinetic description. This generalizes the mono-kinetic, “pressure-less” study by He and Tadmor [Ann. Inst. H. Poincaré C Anal. Non Linéaire 38 (2021), pp. 1031–1053].

We study the (hydro-)dynamics of multi-species driven by alignment. What distinguishes the different species is the protocol of their interaction with the rest of the crowd: the collective motion is described by different communication kernels, ϕ α β , between the crowds in species α and β. We show that flocking of the overall crowd emerges provided the communication array between species forms a connected graph. In particular, the crowd within each species need not interact with its own kind, i.e., ϕ α α = 0 ; different species which are engaged in such ‘game’ of alignment require a connecting path for propagation of information which will lead to the flocking of overall crowd. The same methodology applies to multi-species aggregation dynamics governed by first-order alignment: connectivity implies concentration around an emerging consensus.

We investigate the large time behavior of multi-dimensional aggregation equations driven by Newtonian repulsion, and balanced by radial attraction and confinement. In case of Newton repulsion with radial confinement we quantify the algebraic convergence decay rate toward the unique steady state. To this end, we identify a one-parameter family of radial steady states, and prove dimension-dependent decay rate in energy and 2-Wassertein distance, using a comparison with properly selected radial steady states. We also study Newtonian repulsion and radial attraction. When the attraction potential is quadratic it is known to coincide with quadratic confinement. Here, we study the case of perturbed radial quadratic attraction, proving that it still leads to one-parameter family of unique steady states. It is expected that this family to serve for a corresponding comparison argument which yields algebraic convergence toward steady repulsive-attractive solutions.

We study the multiscale description of large-time collective behavior of agents driven by alignment. The resulting multiflock dynamics arises naturally with realistic initial configurations consist...

We introduce a new class of models for emergent dynamics. It is based on a new communication protocol which incorporates two main features: short-range kernels which restrict the communication to l...

In this paper, we propose the application of third-order semi-discrete central-upwind conservative schemes to option pricing partial differential equations (PDEs). Our method is a high-order extension of the recent efficient second-order “Black-Box” schemes that successfully priced several option pricing problems. We consider the Kurganov–Levy scheme and its extensions, namely the Kurganov–Noelle–Petrova and the Kolb schemes. These “Black-Box” solvers ensure non-oscillatory property and achieve desired accuracy using a third-order central weighted essentially non-oscillatory (CWENO) reconstruction. We compare the schemes using a European test case and observe that the Kolb scheme performs better. We apply the Kolb scheme to one-dimensional butterfly, barrier, American and non-linear options under the Black–Scholes model. Further, we extend the Kurganov–Levy scheme to solve two-dimensional convection-dominated Asian PDE. We also price American options under the constant elasticity of variance (CEV) model, which treats volatility as a stochastic instead of a constant as in Black–Scholes model. Numerical experiments achieve third-order, non-oscillatory and high-resolution solutions.

We overview recent developments in the study of alignment hydrodynamics, driven by a general class of symmetric communication kernels. A main question of interest is to characterize the emergent behavior of such systems, which we quantify in terms of the spectral gap of a weighted Laplacian associated with the alignment operator. Our spectral analysis of energy fluctuation covers both long-range and short-range kernels and does not require thermal equilibrium (no closure for the pressure). In particular, in the prototypical case of metric-based short-range kernels, the spectral gap admits a lower-bound expressed in terms of the discrete Fourier coefficients of the radial kernel, which enables us to quantify an emerging flocking behavior for non-vacuous solutions. These large-time behavior results apply as long as the solutions remain smooth. It is known that global smooth solutions exist in one and two spatial dimensions, subject to sub-critical initial data. We settle the question for arbitrary dimension, obtaining non-trivial initial threshold conditions which guarantee existence of multiD global smooth solutions.

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 continue our study of one-dimensional class of Euler equations, introduced in [ 11 ], driven by a forcing with a commutator structure of the form \begin{document} $[{\mathcal L}_φ, u](ρ)$ \end{document} , where \begin{document} $u$ \end{document} is the velocity field and \begin{document} ${\mathcal L}_φ$ \end{document} belongs to a rather general class of convolution operators depending on interaction kernels \begin{document} $φ$ \end{document} . 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 \begin{document} $φ$ \end{document} 's, and singular \begin{document} $φ(r) = r^{-(1+α)}$ \end{document} of order \begin{document} $α∈ [1, 2)$ \end{document} associated with the action of the fractional Laplacian \begin{document} ${\mathcal L}_φ=-(-\partial_{xx})^{α/2}$ \end{document} . Specifically, we prove fast velocity alignment as the velocity \begin{document} $u(·, t)$ \end{document} approaches a constant state, \begin{document} $u \to \bar{u}$ \end{document} , with exponentially decaying slope and curvature bounds \begin{document} $|{u_x}( \cdot ,t){|_\infty } + |{u_{xx}}( \cdot ,t){|_\infty }\lesssim{e^{ - \delta t}}$ \end{document} . The alignment is accompanied by exponentially fast flocking of the density towards a fixed traveling state \begin{document} $ρ(·, t) -{ρ_{∞}}(x -\bar{u} t) \to 0$ \end{document} .

A standard paradigm for the existence of solutions in fluid dynamics is based on the construction of sequences of approximate solutions or approximate minimizers. This approach faces serious obstacles, most notably in multi-dimensional problems, where the persistence of oscillations at ever finer scales prevents compactness. Indeed, these oscillations are an indication, consistent with recent theoretical results, of the possible lack of existence/uniqueness of solutions within the standard framework of integrable functions. It is in this context that Young measures – parametrized probability measures which can describe the limits of such oscillatory sequences – offer the more general paradigm of measure-valued solutions for these problems.We present viable numerical algorithms to compute approximate measure-valued solutions, based on the realization of approximate measures as laws of Monte Carlo sampled random fields. We prove convergence of these algorithms to measure-valued solutions for the equations of compressible and incompressible inviscid fluid dynamics, and present a large number of numerical experiments which provide convincing evidence for the viability of the new paradigm. We also discuss the use of these algorithms, and their extensions, in uncertainty quantification and contexts other than fluid dynamics, such as non-convex variational problems in materials science.

We investigate the large time behavior of \begin{document}$ N $\end{document} particles restricted to a smooth closed curve in \begin{document}$ \mathbb{R}^d $\end{document} and subject to a gradient flow with respect to Euclidean hyper-singular repulsive Riesz \begin{document}$ s $\end{document}-energy with \begin{document}$ s>1. $\end{document} We show that regardless of their initial positions, for all \begin{document}$ N $\end{document} and time \begin{document}$ t $\end{document} large, their normalized Riesz \begin{document}$ s $\end{document}-energy will be close to the \begin{document}$ N $\end{document}-point minimal possible energy. Furthermore, the distribution of such particles will be close to uniform with respect to arclength measure along the curve.

We study the regularity and large-time behavior of a crowd of species driven by chemo-tactic interactions. What distinguishes the different species is the way they interact with the rest of the crowd: the collective motion is driven by different chemical reactions which end up in a coupled system of parabolic Patlak-Keller-Segel equations. We show that the densities of the different species diffuse to zero provided the chemical interactions between the different species satisfy certain sub-critical condition; the latter is intimately related to a log-Hardy-Littlewood-Sobolev inequality for systems due to Shafrir & Wolansky. Thus for example, when two species interact, one of which has mass less than $4��$, then the 2-system stays smooth for all time independent of the total mass of the system, in sharp contrast with the well-known breakdown of one specie with initial mass$> 8��$.

We study the large-time behavior of systems driven by radial potentials, which react to anticipated positions, $$\mathbf{x}^\tau (t)=\mathbf{x}(t)+\tau \mathbf{v}(t)$$ , with anticipation increment $$\tau >0$$ . As a special case, such systems yield the celebrated Cucker–Smale model for alignment, coupled with pairwise interactions. Viewed from this perspective, such anticipation-driven systems are expected to emerge into flocking due to alignment of velocities, and spatial concentration due to confining potentials. We treat both the discrete dynamics and large crowd hydrodynamics, proving the decisive role of anticipation in driving such systems with attractive potentials into velocity alignment and spatial concentration. We also study the concentration effect near equilibrium for anticipated-based dynamics of pair of agents governed by attractive–repulsive potentials.

We study the large-time behavior of a hydrodynamic model which describes the collective behavior of continuum of agents, driven by pairwise alignment interactions with additional external potential forcing. The external force tends to compete with the alignment which makes the large time behavior very different from the original Cucker–Smale (CS) alignment model, and far more interesting. Here we focus on uniformly convex potentials. In the particular case of quadratic potentials, we are able to treat a large class of admissible interaction kernels, $$\phi (r) > rsim (1+r^2)^{-\beta }$$ with ‘thin’ tails $$\beta \leqslant 1$$ —thinner than the usual ‘fat-tail’ kernels encountered in CS flocking $$\beta \leqslant \nicefrac {1}{2}$$ ; we discover unconditional flocking with exponential convergence of velocities and positions towards a Dirac mass traveling as harmonic oscillator. For general convex potentials, we impose a stability condition, requiring a large enough alignment kernel to avoid crowd scattering. We then prove, by hypocoercivity arguments, that both the velocities and positions of a smooth solution must flock. We also prove the existence of global smooth solutions for one and two space dimensions, subject to critical thresholds in initial configuration space. It is interesting to observe that global smoothness can be guaranteed for sub-critical initial data, independently of the apriori knowledge of large time flocking behavior.

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. &nbsp 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.

We study the large-time behavior of continuum alignment dynamics based on Cucker-Smale (CS)-type interactions which involve short-range kernels, that is, communication kernels with support much smaller than the diameter of the crowd. We show that if the amplitude of the interactions is larger than a finite threshold, then unconditional hydrodynamic flocking follows. Since we do not impose any regularity nor do we require the kernels to be bounded, the result covers both regular and singular interaction kernels. Moreover, we treat initial densities in the general class of compactly supported measures which are required to have positive mass on average (over balls at small enough scale), but otherwise vacuum is allowed at smaller scales. Consequently, our arguments of hydrodynamic flocking apply, mutatis mutandis, to the agent-based CS model with finitely many Dirac masses. In particular, discrete flocking threshold is shown to depend on the number of dense clusters of communication but otherwise does not grow with the number of agents.

The high-order accuracy of Fourier method makes it the method of choice in many large scale simulations. We discuss here the stability of Fourier method for nonlinear evolution problems, focusing on the two prototypical cases of the inviscid Burgers' equation and the multi-dimensional incompressible Euler equations. The Fourier method for such problems with quadratic nonlinearities comes in two main flavors. One is the spectral Fourier method. The other is the $$2/3$$ 2 / 3 pseudo-spectral Fourier method, where one removes the highest $$1/3$$ 1 / 3 portion of the spectrum; this is often the method of choice to maintain the balance of quadratic energy and avoid aliasing errors. Two main themes are discussed in this paper. First, we prove that as long as the underlying exact solution has a minimal $$C^{1+\alpha }$$ C 1 + ? spatial regularity, then both the spectral and the $$2/3$$ 2 / 3 pseudo-spectral Fourier methods are stable. Consequently, we prove their spectral convergence for smooth solutions of the inviscid Burgers equation and the incompressible Euler equations. On the other hand, we prove that after a critical time at which the underlying solution lacks sufficient smoothness, then both the spectral and the $$2/3$$ 2 / 3 pseudo-spectral Fourier methods exhibit nonlinear instabilities which are realized through spurious oscillations. In particular, after shock formation in inviscid Burgers' equation, the total variation of bounded (pseudo-) spectral Fourier solutions must increase with the number of increasing modes and we stipulate the analogous situation occurs with the 3D incompressible Euler equations: the limiting Fourier solution is shown to enforce $$L^2$$ L 2 -energy conservation, and the contrast with energy dissipating Onsager solutions is reflected through spurious oscillations.

We review a general class of models for self-organized dynamics based on alignment. The dynamics of such systems is governed solely by interactions among individuals or “agents,” with the tendency to adjust to their “environmental averages.” This, in turn, leads to the formation of clusters, e.g., colonies of ants, flocks of birds, parties of people, rendezvous in mobile networks, etc. A natural question which arises in this context is to ask when and how clusters emerge through the self-alignment of agents, and what types of “rules of engagement” influence the formation of such clusters. Of particular interest to us are cases in which the self-organized behavior tends to concentrate into one cluster, reflecting a consensus of opinions, flocking of birds, fish, or cells, rendezvous of mobile agents, and, in general, concentration of other traits intrinsic to the dynamics. Many standard models for self-organized dynamics in social, biological, and physical sciences assume that the intensity of alignment in...

We develop a second-order well-balanced central-upwind scheme for the compressible Euler equations with gravitational source term. Here, we advocate a new paradigm based on a purely conservative reformulation of the equations using global fluxes. The proposed scheme is capable of exactly preserving steady-state solutions expressed in terms of a nonlocal equilibrium variable. A crucial step in the construction of the second-order scheme is a well-balanced piecewise linear reconstruction of equilibrium variables combined with a well-balanced central-upwind evolution in time, which is adapted to reduce the amount of numerical viscosity when the flow is at (near) steady-state regime. We show the performance of our newly developed central-upwind scheme and demonstrate importance of perfect balance between the fluxes and gravitational forces in a series of one- and two-dimensional examples.

We study the systems of Euler equations which arise from agent-based dynamics driven by velocity \emph{alignment}. It is known that smooth solutions of such systems must flock, namely -- the large time behavior of the velocity field approaches a limiting "flocking" velocity. To address the question of global regularity, we derive sharp critical thresholds in the phase space of initial configuration which characterize the global regularity and hence flocking behavior of such \emph{two-dimensional} systems. Specifically, we prove for that a large class of \emph{sub-critical} initial conditions such that the initial divergence is "not too negative" and the initial spectral gap is "not too large", global regularity persists for all time.

In this paper, we establish analyticity of the Navier–Stokes equations with small data in critical Besov spaces \({\dot{B}^{\frac{3}{p}-1}_{p,q}}\) . The main method is Gevrey estimates, the choice of which is motivated by the work of Foias and Temam (Contemp Math 208:151–180, 1997). We show that mild solutions are Gevrey regular, that is, the energy bound \({\|e^{\sqrt{t}\Lambda}v(t)\|_{E_p}>\infty}\) holds in \({E_p:=\tilde{L}^{\infty}(0,T;\dot{B}^{\frac{3}{p}-1}_{p,q})\cap \tilde{L}^{1}(0,T;\dot{B}^{\frac{3}{p}+1}_{p,q})}\) , globally in time for p < ∞. We extend these results for the intricate limiting case p = ∞ in a suitably designed E∞ space. As a consequence of analyticity, we obtain decay estimates of weak solutions in Besov spaces. Finally, we provide a regularity criterion in Besov spaces.

We prove stability estimates for the ENO reconstruction and ENO interpolation procedures. In particular, we show that the jump of the reconstructed ENO pointvalues at each cell interface has the same sign as the jump of the underlying cell averages across that interface. We also prove that the jump of the reconstructed values can be upper-bounded in terms of the jump of the underlying cell averages. Similar sign properties hold for the ENO interpolation procedure. These estimates, which are shown to hold for ENO reconstruction and interpolation of arbitrary order of accuracy and on non-uniform meshes, indicate a remarkable rigidity of the piecewise-polynomial ENO procedure.

We design efficient numerical schemes for approximating the MHD equations in multi-dimensions. Numerical approximations must be able to deal with the complex wave structure of the MHD equations and the divergence constraint. We propose schemes based on the genuinely multi-dimensional (GMD) framework of [S. Mishra and E. Tadmor, Commun. Comput. Phys. 9 (2010) 688–710; S. Mishra and E. Tadmor, SIAM J. Numer. Anal. 49 (2011) 1023–1045]. The schemes are formulated in terms of vertex-centered potentials . A suitable choice of the potential results in GMD schemes that preserve a discrete version of divergence. First- and second-order divergence preserving GMD schemes are tested on a series of benchmark numerical experiments. They demonstrate the computational efficiency and robustness of the GMD schemes.

We design arbitrarily high-order accurate entropy stable schemes for systems of conservation laws. The schemes, termed TeCNO schemes, are based on two main ingredients: (i) high-order accurate entropy conservative fluxes and (ii) suitable numerical diffusion operators involving ENO reconstructed cell-interface values of scaled entropy variables. Numerical experiments in one and two space dimensions are presented to illustrate the robust numerical performance of the TeCNO schemes.

Spectral approximations to nonlinear hyperbolic conservation laws require dissipative regularization for stability. The dissipative mechanism must, on the other hand, be small enough in order to retain the spectral accuracy in regions where the solution is smooth. We introduce a new form of viscous regularization which is activated only in the local neighborhood of shock discontinuities. The basic idea is to employ a spectral edge detection algorithm as a dynamical indicator of where in physical space to apply numerical viscosity. The resulting spatially local viscosity is successfully combined with spectral viscosity, where a much higher than usual cut-off frequency can be used. Numerical results show that the new adaptive spectral viscosity scheme significantly improves the accuracy of the standard spectral viscosity scheme. In particular, results are improved near the shocks and at low resolutions. Examples include numerical simulations of Burgers' equation, shallow water with bottom topography, and the isothermal Euler equations. We also test the schemes on a nonconvex scalar problem, finding that the new scheme approximates the entropy solution more reliably than the standard spectral viscosity scheme.

In this paper we present two versions of the central local discontinuous Galerkin (LDG) method on overlapping cells for solving diffusion equations, and provide their stability analysis and error estimates for the linear heat equation. A comparison between the traditional LDG method on a single mesh and the two versions of the central LDG method on overlapping cells is also made. Numerical experiments are provided to validate the quantitative conclusions from the analysis and to support conclusions for general polynomial degrees.

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 consider constraint preserving multidimensional evolution equations. A prototypical example is provided by the magnetic induction equation of plasma physics. The constraint of interest is the divergence of the magnetic field. We design finite volume schemes which approximate these equations in a stable manner and preserve a discrete version of the constraint. The schemes are based on reformulating standard edge centered finite volume fluxes in terms of vertex centered potentials. The potential-based approach provides a general framework for faithful discretizations of constraint transport and we apply it to both divergence preserving as well as curl preserving equations. We present benchmark numerical tests which confirm that our potential-based schemes achieve high resolution, while being constraint preserving.

We introduce a class of numerical schemes that preserve a discrete version of vorticity in conservation laws which involve grad advection. These schemes are based on reformulating finite volume schemes in terms of vertex centered numerical potentials. The resulting potential-based schemes have a genuinely multidimensional structure. A suitable choice of potentials leads to discrete vorticity preserving schemes that are simple to code, computationally inexpensive, and proven to be stable. We extend our discussion to other classes of genuinely multidimensional schemes. Numerical examples for linear grad advection equations, linear and nonlinear wave equation systems, and the Euler equations of gas dynamics are presented.

A novel approach for multiscale image processing based on integro-differential equations (IDEs) was proposed in [E. Tadmor and P. Athavale, Inverse Probl. Imaging, 3 (2009), pp. 693-710]. These IDEs, which stem naturally from multiscale $(BV,L^2)$ hierarchical decompositions, yield inverse scale representations of images in the sense that the $BV$-dual norms of their residuals are inversely proportional to the scaling parameters. Motivated by the fact that $(BV,L^1)$ decomposition is more suitable for extracting local scale-space features than $(BV,L^2)$, we introduce here the IDEs which arise from multiscale $(BV,L^1)$ hierarchical decompositions. We study several variants of this $(BV,L^1)$-based IDE model, depending on modifications to the curvature term.

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 .

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 moti- vation 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.

We propose three novel methods for recovering edges in piecewise smooth functions from their possibly incomplete and noisy spectral information. The proposed methods utilize three different approaches: #1. The randomly-based sparse Inverse Fast Fourier Transform (sIFT); #2. The Total Variation-based (TV) compressed sensing; and #3. The modified zero crossing. The different approaches share a common feature: edges are identified through separation of scales. To this end, we advocate here the use of concentration kernels (Tadmor, Acta Numer. 16:305–378, 2007), to convert the global spectral data into an approximate jump function which is localized in the immediate neighborhoods of the edges. Building on these concentration kernels, we show that the sIFT method, the TV-based compressed sensing and the zero crossing yield effective edge detectors, where finitely many jump discontinuities are accurately recovered. One- and two-dimensional numerical results are presented.

We prove stability and derive error estimates for the recently introduced central discontinuous Galerkin method, in the context of linear hyperbolic equations with possibly discontinuous solutions. A comparison between the central discontinuous Galerkin method and the regular discontinuous Galerkin method in this context is also made. Numerical experiments are provided to validate the quantitative conclusions from the analysis.

We 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 \stackrel{{}_

We prove that the one-dimensional Euler�Poisson system driven by the Poisson forcing together with the usual $\gamma$-law pressure, $\gamma \geq 1$, admits global solutions for a large class of initial data. Thus, the Poisson forcing regularizes the generic finite-time breakdown in the 2x2 p-system. Global regularity is shown to depend on whether or not the initial configuration of the Riemann invariants and density crosses an intrinsic critical threshold.

We prove the finite time blow-up for $C^1$ solutions of the attractive Euler-Poisson equations in $\Bbb R^{2}$, $n\geq1$, with and without background state, for a large set of ’generic’ initial data. We characterize this supercritical set by tracing the spectral dynamics of the deformation and vorticity tensors.

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 $\sqrt{\eta}$, and the $\mathcal{O}(1)$ scale of the jump discontinuities. We use concentration factors which are adjusted to the standard deviation of the noise $\sqrt{\eta} \gg 1/N$ in order to detect the underlying $\mathcal{O}(1)$-edges, which are separated from the noise scale $\sqrt{\eta} \ll 1$.