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 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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Giuseppe Maria Coclite, Jean-Michel Coron, Nicola De Nitti, Alexander Keimer, Lukas Pflug, A general result on the approximation of local conservation laws by nonlocal conservation laws: The singular limit problem for exponential kernels. Ann. Inst. H. Poincaré Anal. Non Linéaire 40 (2023), no. 5, pp. 1205–1223DOI 10.4171/AIHPC/58