On the role of numerical viscosity in the study of the local limit of nonlocal conservation laws

Nonlocal conservation laws (the signature feature being that the flux function depends on the solution through the convolution with a given kernel) are extensively used in the modeling of vehicular traffic. In this work we discuss the singular local limit, namely the convergence of the nonlocal solutions to the entropy admissible solution of the conservation law obtained by replacing the convolution kernel with a Dirac delta. Albeit recent counter-examples rule out convergence in the general case, in the specific framework of traffic models (with anisotropic convolution kernels) the singular limit has been established under rigid assumptions, i.e. in the case of the exponential kernel (which entails algebraic identities between the kernel and its derivatives) or under fairly restrictive requirements on the initial datum. In this work we obtain general convergence results under assumptions that are entirely natural in view of applications to traffic models, plus a convexity requirement on the convolution kernels. We then provide a general criterion for entropy admissibility of the limit and a convergence rate. We also exhibit a counter-example showing that the convexity assumption is necessary for our main compactness estimate.

Local limit of nonlocal traffic models: Convergence results and total variation blow-up

We give an answer to a question posed in Amorim et al. (ESAIM Math Model Numer Anal 49(1):19–37, 2015), which can loosely speaking, be formulated as follows: consider a family of continuity equations where the velocity depends on the solution via the convolution by a regular kernel. In the singular limit where the convolution kernel is replaced by a Dirac delta, one formally recovers a conservation law. Can we rigorously justify this formal limit? We exhibit counterexamples showing that, despite numerical evidence suggesting a positive answer, one does not in general have convergence of the solutions. We also show that the answer is positive if we consider viscous perturbations of the nonlocal equations. In this case, in the singular local limit the solutions converge to the solution of the viscous conservation law.

Foreword.- Preface.- Open questions in the theory of one dimensional hyperbolic conservation laws.- Multidimensional conservation laws: Overview, problems, and perspective.- Mathematical analysis of fluids in motion.- Selected topics in approximate solutions of nonlinear conservation laws.- High-resolution central schemes.- Stability and dynamics of viscous shock waves.- Mathematical aspects of a model for granular flow.- The flow associated to weakly differentiable vector fields: recent results and open problems.- Existence and uniqueness results for the continuity equation and applications to the chromatography system.- Finite energy weak solutions to the quantum hydrodynamics system.- The Monge problem in geodesic spaces.- Existence of a unique solution to a nonlinear moving-boundary problem of mixed type arising in modeling blood flow.- Transonic flows and isometric embeddings.- Well posedness and control in models based on conservation laws.-Homogenization of nonlinear partial differential equations in the context of ergodic algebras: Recent results and open problems.- Conservation laws at a node.- Nonlinear hyperbolic surface waves.- Vacuum in gas and fluid dynamics.- On radially symmetric solutions to conservation laws.- Charge transport in an incompressible fluid: New devices in computational electronics.- Localization and shear bands in high strain-rate plasticity.- Hyperbolic conservation laws on spacetimes.- Reduced theories in nonlinear elasticity.- Mathematical, physical and numerical principles essential for models of turbulent mixing.- On the Euler-Poisson equations of self-gravitating compressible fluids.- Viscous system of conservation laws: Singular limits.- A two-dimensional Riemann problem for scalar conservation laws.- Semi-hyperbolic waves in two-dimensional compressible Euler systems.- List of summer program participants.

Abstract. The propagation of long, weakly nonlinear internal waves in a stratified gas is studied. Hydrodynamic equations for an ideal fluid with the perfect gas law describe the atmospheric gas behaviour. If we neglect the term Ͽ dw/dt (product of the density and vertical acceleration), we come to a so-called quasistatic model, while we name the full hydro-dynamic model as a nonquasistatic one. Both quasistatic and nonquasistatic models are used for wave simulation and the models are compared among themselves. It is shown that a smooth classical solution of a nonlinear quasistatic problem does not exist for all t because a gradient catastrophe of non-linear internal waves occurs. To overcome this difficulty, we search for the solution of the quasistatic problem in terms of a generalised function theory as a limit of special regularised equations containing some additional dissipation term when the dissipation factor vanishes. It is shown that such solutions of the quasistatic problem qualitatively differ from solutions of a nonquasistatic nature. It is explained by the fact that in a nonquasistatic model the vertical acceleration term plays the role of a regularizator with respect to a quasistatic model, while the solution qualitatively depends on the regularizator used. The numerical models are compared with some analytical results. Within the framework of the analytical model, any internal wave is described as a system of wave modes; each wave mode interacts with others due to equation non-linearity. In the principal order of a perturbation theory, each wave mode is described by some equation of a KdV type. The analytical model reveals that, in a nonquasistatic model, an internal wave should disintegrate into solitons. The time of wave disintegration into solitons, the scales and amount of solitons generated are important characteristics of the non-linear process; they are found with the help of analytical and numerical investigations. Satisfactory coincidence of simulation outcomes with analytical ones is revealed and some examples of numerical simulations illustrating wave disintegration into solitons are given. The phenomenon of internal wave mixing is considered and is explained from the point of view of the results obtained. The numerical methods for internal wave simulation are examined. In particular, the influence of difference interval finiteness on a numerical solution is investigated. It is revealed that a numerical viscosity and numerical dispersion can play the role of regularizators to a nonlinear quasistatic problem. To avoid this effect, the grid steps should be taken less than some threshold values found theoretically.

We study a rather general class of 1D nonlocal conservation laws from a numerical point of view. First, following [F. Betancourt, R. Bürger, K.H. Karlsen and E.M. Tory, On nonlocal conservation laws modelling sedimentation. Nonlinearity 24 (2011) 855–885], we define an algorithm to numerically integrate them and prove its convergence. Then, we use this algorithm to investigate various analytical properties, obtaining evidence that usual properties of standard conservation laws fail in the nonlocal setting. Moreover, on the basis of our numerical integrations, we are led to conjecture the convergence of the nonlocal equation to the local ones, although no analytical results are, to our knowledge, available in this context.

New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection–Diffusion Equations

Singular limits with vanishing viscosity for nonlocal conservation laws

We deal with the problem of approximating a scalar conservation law by a conservation law with nonlocal flux. As convolution kernel in the nonlocal flux, we consider an exponential-type approximation of the Dirac distribution. We then obtain a total variation bound on the nonlocal term and can prove that the (unique) weak solution of the nonlocal problem converges strongly in C({L_{loc}^{1}}) to the entropy solution of the local conservation law. We conclude with several numerical illustrations which underline the main results and, in particular, the difference between the solution and the nonlocal term.

On approximation of local conservation laws by nonlocal conservation laws

CHAPTER 4 - LINEAR MOMENTUM

We consider a general class of convolution-type nonlocal wave equations modeling bidirectional nonlinear wave propagation. The model involves two small positive parameters measuring the relative strengths of the nonlinear and dispersive effects. We take two different kernel functions that have similar dispersive characteristics in the long-wave limit and compare the corresponding solutions of the Cauchy problems with the same initial data. We prove rigorously that the difference between the two solutions remains small over a long time interval in a suitable Sobolev norm. In particular, our results show that, in the long-wave limit, solutions of such nonlocal equations can be well approximated by those of improved Boussinesq-type equations.

Partial differential equations of the form \documentclass[12pt]{minimal}\begin{document}$\mathop {\hbox{\rm div}}{{\bm N}} =0$\end{document}divN=0, \documentclass[12pt]{minimal}\begin{document}${{\bm N}}_t + \mathop {\hbox{\rm curl}}{{\bm M}}=0$\end{document}Nt+curlM=0 involving two vector functions in \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^3$\end{document}R3 depending on t, x, y, z appear in different physical contexts, including the vorticity formulation of fluid dynamics, magnetohydrodynamics (MHD) equations, and Maxwell's equations. It is shown that these equations possess an infinite family of local divergence-type conservation laws involving arbitrary functions of space and time. Moreover, it is demonstrated that the equations of interest have a rather special structure of a lower-degree (degree two) conservation law in \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^4(t,x,y,z)$\end{document}R4(t,x,y,z). The corresponding potential system has a clear physical meaning. For the Maxwell's equations, it gives rise to the scalar electric and the vector magnetic potentials; for the vorticity equations of fluid dynamics, the potentialization inverts the curl operator to yield the fluid dynamics equations in primitive variables; for MHD equations, the potential equations yield a generalization of the Galas-Bogoyavlenskij potential that describes magnetic surfaces of ideal MHD equilibria. The lower-degree conservation law is further shown to yield curl-type conservation laws and determined potential equations in certain lower-dimensional settings. Examples of new nonlocal conservation laws, including an infinite family of nonlocal material conservation laws of ideal time-dependent MHD equations in 2+1 dimensions, are presented.

We consider the numerical solution of nonlocal constrained value problems associated with linear nonlocal diffusion and nonlocal peridynamic models. Two classes of discretization methods are presented, including standard finite element methods and quadrature-based finite difference methods. We discuss the applicability of these approaches to nonlocal problems having various singular kernels and study basic numerical analysis issues. We illustrate the similarities and differences of the resulting nonlocal stiffness matrices and discuss whether discrete maximum principles can be established. We pay particular attention to the issue of convergence in both the nonlocal setting and the local limit. While it is known that the nonlocal models converge to corresponding differential equations in the local limit, we elucidate how such limiting behaviors may or may not be preserved in various discrete approximations. Our findings thus offer important insight into applications and simulations of nonlocal models.

In this paper we consider 2D nonlocal diffusion models with a finite nonlocal horizon parameterδcharacterizing the range of nonlocal interactions, and consider the treatment of Neumann-like boundary conditions that have proven challenging for discretizations of nonlocal models. We propose a new generalization of classical local Neumann conditions by converting the local flux to a correction term in the nonlocal model, which provides an estimate for the nonlocal interactions of each point with points outside the domain. While existing 2D nonlocal flux boundary conditions have been shown to exhibit at most first order convergence to the local counter part asδ → 0, the proposed Neumann-type boundary formulation recovers the local case asO(δ2) in theL∞(Ω) norm, which is optimal considering theO(δ2) convergence of the nonlocal equation to its local limit away from the boundary. We analyze the application of this new boundary treatment to the nonlocal diffusion problem, and present conditions under which the solution of the nonlocal boundary value problem converges to the solution of the corresponding local Neumann problem as the horizon is reduced. To demonstrate the applicability of this nonlocal flux boundary condition to more complicated scenarios, we extend the approach to less regular domains, numerically verifying that we preserve second-order convergence for non-convex domains with corners. Based on the new formulation for nonlocal boundary condition, we develop an asymptotically compatible meshfree discretization, obtaining a solution to the nonlocal diffusion equation with mixed boundary conditions that converges withO(δ2) convergence.