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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 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 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...

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 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.

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 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 $\sqrt{\eta} \ll 1$.

We are concerned here with processing discontinuous functions from their spectral information. We focus on two main aspects of processing such piecewise smooth data: detecting the edges of a piecewise smooth f, namely, the location and amplitudes of its discontinuities; and recovering with high accuracy the underlying function in between those edges. If f is a smooth function, say analytic, then classical Fourier projections recover f with exponential accuracy. However, if f contains one or more discontinuities, its global Fourier projections produce spurious Gibbs oscillations which spread throughout the smooth regions, enforcing local loss of resolution and global loss of accuracy. Our aim in the computation of the Gibbs phenomenon is to detect edges and to reconstruct piecewise smooth functions, while regaining the high accuracy encoded in the spectral data.To detect edges, we utilize a general family of edge detectors based on concentration kernels. Each kernel forms an approximate derivative of the delta function, which detects edges by separation of scales. We show how such kernels can be adapted to detect edges with one- and two-dimensional discrete data, with noisy data, and with incomplete spectral information. The main feature is concentration kernels which enable us to convert global spectral moments into local information in physical space. To reconstruct f with high accuracy we discuss novel families of mollifiers and filters. The main feature here is making these mollifiers and filters adapted to the local region of smoothness while increasing their accuracy together with the dimension of the data. These mollifiers and filters form approximate delta functions which are properly parametrized to recover f with (root-) exponential accuracy.

We prove in this paper new velocity-averaging results for second-order multidimensional equations of the general form L(∇x ,v )f (x ,v ) = g(x ,v )where L(∇x ,v ):= a(v) ·∇ x −∇ � ·b(v)∇x. These results quantify the Sobolev regularity of the averages, � v f (x, v)φ(v)dv, in terms of the nondegeneracy of the set

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.

We are concerned with the detection of edges--the location and amplitudes of jump discontinuities of piecewise smooth data realized in terms of its discrete grid values. We discuss the interplay between two approaches. One approach, realized in the physical space, is based on local differences and is typically limited to low-order of accuracy. An alternative approach developed in our previous work [Gelb and Tadmor, Appl. Comp. Harmonic Anal., 7, 101---135 (1999)] and realized in the dual Fourier space, is based on concentration factors; with a proper choice of concentration factors one can achieve higher-orders--in fact in [Gelb and Tadmor, SIAM J. Numer. Anal., 38, 1389---1408 (2001)] we constructed exponentially accurate edge detectors. Since the stencil of these highly-accurate detectors is global, an outside threshold parameter is required to avoid oscillations in the immediate neighborhood of discontinuities. In this paper we introduce an adaptive edge detection procedure based on a cross-breading between the local and global detectors. This is achieved by using the minmod limiter to suppress spurious oscillations near discontinuities while retaining high-order accuracy away from the jumps. The resulting method provides a family of robust, parameter-free edge-detectors for piecewise smooth data. We conclude with a series of one- and two-dimensional simulations.

We present a new family of high-resolution, nonoscillatory semidiscrete central schemes for the approximate solution of the ideal magnetohydrodynamics (MHD) equations. This is the second part of our work, where we are passing from the fully discrete staggered schemes in [J. Balbas, E. Tadmor, and C.-C. Wu, J. Comput. Phys., 201 (2004), pp. 261-285] to the semidiscrete formulation advocated in [A. Kurganov and E. Tadmor, J. Comput. Phys., 160 (2000), pp. 241-282]. This semidiscrete formulation retains the simplicity of fully discrete central schemes while enhancing efficiency and adding versatility. The semidiscrete algorithm offers a wider range of options to implement its two key steps: nonoscillatory reconstruction of point values followed by the evolution of the corresponding point valued fluxes. We present the solution of several prototype MHD problems. Solutions of one-dimensional Brio--Wu shock-tube problems and the two-dimensional Kelvin--Helmholtz instability, Orszag--Tang vortex system, and the disruption of a high density cloud by a strong shock are carried out using third- and fourth-order central schemes based on the central WENO reconstructions. These results complement those presented in our earlier work and confirm the remarkable versatility and simplicity of central schemes as black-box, Jacobian-free MHD solvers. Furthermore, our numerical experiments demonstrate that this family of semidiscrete central schemes preserves the $\nabla \cdot {\bf B} = 0$-constraint within machine round-off error; happily, no constrained-transport enforcement is needed.

We introduce a new class of exponentially accurate filters for processing piecewise smooth spectral data. Our study is based on careful error decompositions, focusing on a rather precise balance between physical space localization and the usual moments condition. Exponential convergence is recovered by optimizing the order of the filter as an adaptive function of both the projection order and the distance to the nearest discontinuity. Combined with the automated edge detection methods, e.g. Gelb & Tadmor (2002, Math. Model. Numer. Anal., 36, 155–175), adaptive filters provide a robust, computationally efficient, black box procedure for the exponentially accurate reconstruction of a piecewise smooth function from its spectral information.

We study the entropy stability of difference approximations to nonlinear hyperbolic conservation laws, and related time-dependent problems governed by additional dissipative and dispersive forcing terms. We employ a comparison principle as the main tool for entropy stability analysis, comparing the entropy production of a given scheme against properly chosen entropy-conservative schemes.To this end, we introduce general families of entropy-conservative schemes, interesting in their own right. The present treatment of such schemes extends our earlier recipe for construction of entropy-conservative schemes, introduced in Tadmor (1987b). The new families of entropy-conservative schemes offer two main advantages, namely, (i) their numerical fluxes admit an explicit, closed-form expression, and (ii) by a proper choice of their path of integration in phase space, we can distinguish between different families of waves within the same computational cell; in particular, entropy stability can be enforced on rarefactions while keeping the sharp resolution of shock discontinuities.A comparison with the numerical viscosities associated with entropy-conservative schemes provides a useful framework for the construction and analysis of entropy-stable schemes. We employ this framework for a detailed study of entropy stability for a host of first- and second-order accurate schemes. The comparison approach yields a precise characterization of the entropy stability of semi-discrete schemes for both scalar problems and systems of equations.We extend these results to fully discrete schemes. Here, spatial entropy dissipation is balanced by the entropy production due to time discretization with a suffciently small time-step, satisfying a suitable CFL condition. Finally, we revisit the question of entropy stability for fully discrete schemes using a different approach based on homotopy arguments. We prove entropy stability under optimal CFL conditions.

We report here on our numerical study of the two-dimensional Riemann problem for the com- pressible Euler equations. Compared with the relatively simple 1-D congurations, the 2-D case consists of a plethora of geometric wave patterns which pose a computational challenge for high- resolution methods. The main feature in the present computations of these 2-D waves is the use of the Riemann-solvers-free central schemes presented in (11). This family of central schemes avoids the intricate and time-consuming computation of the eigensystem of the problem, and hence oers a considerably simpler alternative to upwind methods. The numerical results illustrate that despite their simplicity, the central schemes are able to recover with comparable high-resolution, the various features observed in the earlier, more expensive computations. AMS subject classication: Primary 65M10; Secondary 65M05

We discuss the reconstruction of piecewise smooth data from its (pseudo-) spectral information. Spectral projections enjoy superior resolution provided the data is globally smooth, while the presence of jump discontinuities is responsible for spurious ${\cal O}(1)$ Gibbs oscillations in the neighborhood of edges and an overall deterioration to the unacceptable first-order convergence rate. The purpose is to regain the superior accuracy in the piecewise smooth case, and this is achieved by mollification. Here we utilize a modified version of the two-parameter family of spectral mollifiers introduced by Gottlieb & Tadmor [GoTa85]. The ubiquitous one-parameter, finite-order mollifiers are based on dilation. In contrast, our mollifiers achieve their high resolution by an intricate process of high-order cancelation. To this end, we first implement a localization step using edge detection procedure, [GeTa00a, GeTa00b]. The accurate recovery of piecewise smooth data is then carried out in the direction of smoothness away from the edges, and adaptivity is responsible for the high resolution. The resulting adaptive mollifier greatly accelerates the convergence rate, recovering piecewise analytic data within exponential accuracy while removing spurious oscillations that remained in [GoTa85]. Thus, these adaptive mollifiers offer a robust, general-purpose ``black box'' procedure for accurate post processing of piecewise smooth data.

In 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. We 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. We show the exact blow-up rate when the slope of the initial profile is below the lower threshold. We further prove the L1 stability of the smooth shock profile, provided the slope of the initial profile is above the critical threshold.

In this paper we review and further develop a class of strong stability-preserving (SSP) high-order time discretizations for semidiscrete method of lines approximations of partial differential equations. Previously termed TVD (total variation diminishing) time discretizations, these high-order time discretization methods preserve the strong stability properties of first-order Euler time stepping and have proved very useful, especially in solving hyperbolic partial differential equations. The new developments in this paper include the construction of optimal explicit SSP linear Runge--Kutta methods, their application to the strong stability of coercive approximations, a systematic study of explicit SSP multistep methods for nonlinear problems, and the study of the SSP property of implicit Runge--Kutta and multistep methods.

We 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 entropy solution under appropriate reasonable conditions. The proposed SV scheme is implemented directly on high modes of the computed solution. This should be compared with the original nonperiodic SV scheme introduced by Maday, 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 by our numerical experiments. A postprocessing method is implemented to greatly enhance the accuracy of the computed SV solution. The numerical results show the efficiency of the new method.

We discuss a general framework for recovering edges in piecewise smooth functions with finitely many jump discontinuities, where $[f](x):=f(x+)-f(x-) \neq 0$. 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, $K_\epsilon(\cdot)$, depending on the small scale $\epsilon$. It is shown that odd kernels, properly scaled, and admissible (in the sense of having small $W^{-1,\infty}$-moments of order ${\cal O}(\epsilon)$) satisfy $K_\epsilon*f(x) = [f](x) +{\cal O}(\epsilon)$, thus recovering both the location and amplitudes of all edges. As an example we consider general concentration kernels of the form $K^\sigma_N(t)=\sum\sigma(k/N)\sin kt$ to detect edges from the first $1/\epsilon=N$ 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, $\sigma^{exp}(\cdot)$, 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 $K_\epsilon*f(x)\sim [f](x) \neq 0$, and the smooth regions where $K_\epsilon*f = {\cal O}(\epsilon) \sim 0$. Numerical examples demonstrate that by coupling concentration kernels with nonlinear enhancement one arrives at effective edge detectors.

We 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.

We construct, analyze, and implement a new nonoscillatory high-resolution scheme for two-dimensional hyperbolic conservation laws. The scheme is a predictor-corrector method which consists of two steps: starting with given cell averages, we first predict pointvalues which are based on nonoscillatory piecewise-linear reconstructions from the given cell averages; at the second corrector step, we use staggered averaging, together with the predicted midvalues, to realize the evolution of these averages. This results in a second-order, nonoscillatory central scheme, a natural extension of the one-dimensional second-order central scheme of Nessyahu and Tadmor [J. Comput. Phys., 87 (1990), pp. 408--448]. As in the one-dimensional case, the main feature of our two-dimensional scheme is simplicity. In particular, this central scheme does not require the intricate and time-consuming (approximate) Riemann solvers which are essential for the high-resolution upwind schemes; in fact, even the computation of the exact Jacobians can be avoided. Moreover, the central scheme is "genuinely multidimensional" in the sense that it does not necessitate dimensional splitting. We prove that the scheme satisfies the scalar maximum principle, and in the more general context of systems, our proof indicates that the scheme is positive (in the sense of Lax and Liu [CFD Journal, 5 (1996), pp. 1--24]). We demonstrate the application of our central scheme to several prototype two-dimensional Euler problems. Our numerical experiments include the resolution of shocks oblique to the computational grid; they show how our central scheme solves with high resolution the intricate wave interactions in the so-called double Mach reflection problem [J. Comput. Phys., 54 (1988), pp. 115--173] without following the characteristics; and finally we report on the accurate ray solutions of a weakly hyperbolic system [J. Comput. Appl. Math., 74 (1996), pp. 175--192], rays which otherwise are missed by the dimensional splitting approach. Thus, a considerable amount of simplicity and robustness is gained while achieving stability and high resolution.

A third-order accurate Godunov-type scheme for the approximate solution of hyperbolic systems of conservation laws is presented. Its two main ingredients include: 1. A non-oscillatory piecewise-quadratic reconstruction of pointvalues from their given cell averages; and 2. A central differencing based on staggered evolution of the reconstructed cell averages. This results in a third-order central scheme, an extension along the lines of the second-order central scheme of Nessyahu and Tadmor \cite{NT}. The scalar scheme is non-oscillatory (and hence – convergent), in the sense that it does not increase the number of initial extrema (– as does the exact entropy solution operator). Extension to systems is carried out by componentwise application of the scalar framework. In particular, we have the advantage that, unlike upwind schemes, no (approximate) Riemann solvers, field-by-field characteristic decompositions, etc., are required. Numerical experiments confirm the high-resolution content of the proposed scheme. Thus, a considerable amount of simplicity and robustness is gained while retaining the expected third-order resolution.

The integration of semidiscrete approximations for time-dependent problems is encountered in a variety of applications. The Runge--Kutta (RK) methods are widely used to integrate the ODE systems which arise in this context, resulting in large ODE systems called methods of lines. These methods of lines are governed by possibly ill-conditioned systems with a growing dimension; consequently, the naive spectral stability analysis based on scalar eigenvalues arguments may be misleading. Instead, we present here a stability analysis of RK methods for well-posed semidiscrete approximations, based on a general energy method. We review the stability question for such RK approximations, and highlight its intricate dependence on the growing dimension of the problem. In particular, we prove the strong stability of general fully discrete RK methods governed by coercive approximations. We conclude with two nontrivial examples which demonstrate the versatility of our approach in the context of general systems of convection-diffusion equations with variable coefficients. A straightforward implementation of our results verify the strong stability of RK methods for local finite-difference schemes as well as global spectral approximations. Since our approach is based on the energy method (which is carried in the physical space), and since it avoids the von Neumann analysis (which is carried in the dual Fourier space), we are able to easily adapt additional extensions due to nonperiodic boundary conditions, general geometries, etc.

We are concerned with $2\times2$ nonlinear relaxation systems of conservation laws of the form $u_t+f(u)_x=-\frac{1}{\delta}S(u,v), v_t=\frac{1}{\delta}S(u,v)$ which are coupled through the stiff source term $\frac{1}{\delta}S(u,v)$. Such systems arise as prototype models for combustion, adsorption, etc. Here we study the convergence of $(u,v)\equiv(\ud,\vd)$ to its equilibrium state, $(\bar{u},\bar{v})$, governed by the limiting equations, $\bar{u}_t+\bar{v}_t+ f(\bar{u})_x=0, S(\bar{u},\bar{v})=0$. In particular, we provide sharp convergence rate estimates as the relaxation parameter $\delta \downarrow 0$. The novelty of our approach is the use of a weak $W^{-1}(L^1)$-measure of the error, which allows us to obtain sharp error estimates. It is shown that the error consists of an initial contribution of size ${||S(u_0^\delta,v_0^\delta)||}_{L^1}$, together with accumulated relaxation error of order ${\cal O}(\delta)$. The sharpness of our results is found to be in complete agreement with the numerical ex...

We consider the 2 x 2 hyperbolic system of isentropic gas dynamics, in both Eulerian or Lagrangian variables (also called the p-system). We show that they can be reformulated as a kinetic equation, using an additional kinetic variable. Such a formulation was first obtained by the authors in the case of multidimensio nal scalar conservation laws. A new phenomenon occurs here, namely that the advection velocity is now a combination of the macroscopic and kinetic velocities. Various applications are given: we recover the invariant regions, deduce new L°° estimates using moments lemma and prove L°° — w* stability for 7 > 3.

It has been a long open question whether the pseudospectral Fourier method without smoothing is stable for hyperbolic equations with variable coefficients that change signs. In this work we answer this question with a detailed stability analysis of prototype cases of the Fourier method. We show that due to weighted \(L^2\)-stability, the \(N\)-degree Fourier solution is algebraically stable in the sense that its \(L^2\) amplification does not exceed \({O}(N)\). Yet, the Fourier method is weakly \(L^2\) -unstable in the sense that it does experience such \({O}(N)\) amplification. The exact mechanism of this weak instability is due the aliasing phenomenon, which is responsible for an \({O}(N)\) amplification of the Fourier modes at the boundaries of the computed spectrum. Two practical conclusions emerge from our discussion. First, the Fourier method is required to have sufficiently many modes in order to resolve the underlying phenomenon. Otherwise, the lack of resolution will excite the weak instability which will propagate from the slowly decaying high modes to the lower ones. Second -- independent of whether smoothing was used or not, the small scale information contained in the highest modes of the Fourier solution will be destroyed by their \({O}(N)\) amplification. Happily, with enough resolution nothing worse can happen.

We present a new formulation of multidimensional scalar conservation laws, which includes both the equation and the entropy criterion. This formulation is a kinetic one involving an additional variable called velocity by analogy. We also give some applications of this formulation to new compactness and regularity results for entropy solutions based upon the velocity-averaging lemmas. Finally, we show that this kinetic formulation is in fact valid and meaningful for more general classes of equations like equations involving nonlinear second-order terms.

This Special Issue is based on research presented at the Workshop ``Modeling and Control of Social Dynamics", hosted by the Center of Computational and Integrative Biology and the Department of Mathematical Sciences at Rutgers University - Camden. The Workshop is part of the activities of the NSF Research Network in Mathematical Sciences: ``Kinetic description of emerging challenges in multiscale problems of natural sciences" Grant # 1107444, which is also acknowledged for funding the workshop. For more information please click the “Full Text” above.

The behavior and structure of entropy solutions of scalar convex conservation laws are studied. It is well kn0a.n that such entropy solutioris consist of at most countable number of C1-smooth regions. We obtain new upper. bounds on the higher order derivatives of the entropy solution in any one of its C1-smoothness regions. These bounds enable us to measure the hzgh order piecewise smoothness of the entropy solution. To this end we introduce an appropriate new Cn-semi norm - localized to the smooth part of the entropy solution. and we show that the entropy solution is stable with respect to this norm. \Ye also address the question regarding the number of C1-smoothness pieces, we show that if the initial speed has a finite number of decreasing inflection points then it bounds the number of future shock discontinuities. Loosely speaking this says that in the case of such generic initial data the entropy solution consists of a finite number of smooth pieces, each of which is as smooth as the data permits. It is this type of pzecewise smoothness which is assumed - sometime implicitly - in many finite-dimensional computations for such discontinuous problems.

We study the behavior of spectral viscosity approximations to non-linear scalar conservation laws. We show how the spectral viscosity method compromises between the total-variation bounded viscosity approximations— which are restricted to first-order accuracy—and the spectrally accurate, yet unstable, Fourier method. In particular, we prove that the spectral viscosity method is L 1 {L^1} -stable and hence total-variation bounded. Moreover, the spectral viscosity solutions are shown to be Lip + {\text {Lip}^ + } -stable, in agreement with Oleinik’s E-entropy condition. This essentially nonoscillatory behavior of the spectral viscosity method implies convergence to the exact entropy solution, and we provide convergence rate estimates of both global and local types.

We study the spectral viscosity (SV) method in the context of multidimensional scalar conservation laws with periodic boundary conditions. We show that the spectral viscosity, which is sufficiently small to retain the formal spectral accuracy of the underlying Fourier approximation, is large enough to enforce the correct amount of entropy dissipation (which is otherwise missing in the standard Fourier method). Moreover, we prove that because of the presence of the spectral viscosity, the truncation error in this case becomes spectrally small, independent of whether the underlying solution is smooth or not. Consequently, the SV approximation remains uniformly bounded and converges to a measure-valued solution satisfying the entropy condition, that is, the unique entropy solution. We also show that the SV solution has a bounded total variation, provided that the total variation of the initial data is bounded, thus confirming its strong convergence to the entropy solution. We obtain an L 1 {L^1} convergence rate of the usual optimal order one-half.

Rosenau has recently proposed a regularized version of the Chapman-Enskog expansion of hydrodynamics. This regularized expansion resembles the usual Navier-Stokes viscosity terms at law wave-numbers, but unlike the latter, it has the advantage of being a bounded macroscopic approximation to the linearized collision operator. The behavior of Rosenau regularization of the Chapman-Enskog expansion (RCE) is studied in the context of scalar conservation laws. It is shown that thie RCE model retains the essential properties of the usual viscosity approximation, e.g., existence of traveling waves, monotonicity, upper-Lipschitz continuity..., and at the same time, it sharpens the standard viscous shock layers. It is proved that the regularized RCE approximation converges to the underlying inviscid entropy solution as its mean-free-path epsilon approaches 0, and the convergence rate is estimated.

We study the stability of spectral approximations to scalar hyperbolic initial-boundary value problems with variable coefficients. Time is discretized by explicit multi-level or Runge-Kutta methods of order ≤ 3 \leq 3 (forward Euler time-differencing is included), and we study spatial discretizations by spectral and pseudospectral approximations associated with the general family of Jacobi polynomials. We prove that these fully explicit spectral approximations are stable provided their time step, Δ t \Delta t , is restricted by the CFL-like condition Δ t > Const ∙ N − 2 \Delta t > {\text {Const}} \bullet {N^{ - 2}} , where N equals the spatial number of degrees of freedom. We give two independent proofs of this result, depending on two different choices of appropriate L 2 {L^2} -weighted norms. In both approaches, the proofs hinge on a certain inverse inequality interesting for its own sake. Our result confirms the commonly held belief that the above CFL stability restriction, which is extensively used in practical implementations, guarantees the stability (and hence the convergence) of fully-explicit spectral approximations in the nonperiodic case.

Consider a scalar, nonlinear conservative difference scheme satisfying the entropy condition. It is shown that difference schemes containing more numerical viscosity will necessarily converge to the unique, physically relevant weak solution of the approximated conservative equation. In particular, entropy satisfying convergence follows for E schemes—those containing more numerical viscosity than Godunov’s scheme.