A general result on the approximation of local conservation laws by nonlocal conservation laws: The singular limit problem for exponential kernels

Abstract

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. This enables us to obtain a total variation bound on the nonlocal term. By using this, we prove that the (unique) weak solution of the nonlocal problem converges strongly in C(L 1 loc ) 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.

Discontinuous nonlocal conservation laws and related discontinuous ODEs -- Existence, Uniqueness, Stability and Regularity

“…From the perspective of approximating local conservation laws by nonlocal conservation laws [14,19,31,40,42,43], we consider the nonlocal approximations of the following discontinuous (local) conservation laws:…”

“…-obsolete (compare with [10]) although still used in literature [54,55]. The established theory sets the stage for several future directions: 1) similar to [40], consideration of the convergence to the local discontinuous conservation law when we let the convolution kernel in the nonlocal part of the velocity converge to a Dirac distribution, 2) the bounded domain case similar to [1], 3) measurevalued solutions similar to [46], assuming that the kernel is in W 1,∞ (R), 4) discontinuous (in space) multi-dimensional nonlocal conservation laws.…”

Network models for nonlocal traffic flow

“…This nonlocal range can stand for the connection radius of autonomous cars or for the sight of a driver. Nonlocal models for traffic flow are widely studied in current research concerning existence of solutions [4,11,24,29,36], numerical schemes [4,8,23,24,29] or convergence to local conservation laws [5,6,15,18,38] -even, in general, this question is still an open problem. Modeling approaches include microscopic models [10,13,28,45], second order models [10], multiclass models [12], multilane models [3,22] and also time delay models [37].…”