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

created by denitti on 24 Dec 2020
modified on 25 Dec 2020

[BibTeX]

Preprint

Inserted: 24 dec 2020
Last Updated: 25 dec 2020

Year: 2020

ArXiv: 2012.13203 PDF

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 $C(L^{1}_{\text{loc}})$ 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.

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