Second-Order Accurate TVD Numerical Methods for Nonlocal Nonlinear Conservation Laws
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2021
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Journal Article
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Abstract
We present a second-order accurate numerical method for a class of nonlocal nonlinear conservation laws called the "nonlocal pair-interaction model," which was recently introduced by Du, Huang, and LeFloch [SIAM J. Numer. Anal., 55 (2017), pp. 2465-2489]. Our numerical method uses second-order accurate reconstruction-based schemes for local conservation laws in conjunction with appropriate numerical integration. We show that the resulting method is total variation diminishing (TVD) and converges towards a weak solution. In fact, in contrast to local conservation laws, our second-order reconstruction-based method converges towards the unique entropy solution provided that the nonlocal interaction kernel satisfies a certain growth condition near zero. Furthermore, as the nonlocal horizon parameter in our method approaches zero we recover a well-known second-order method for local conservation laws. In addition, we answer several questions from the paper by Du, Huang, and LeFloch [SIAM J. Numer. Anal., 55 (2017), pp. 2465-2489] concerning regularity of solutions. In particular, we prove that any discontinuity present in a weak solution must be stationary and that if the interaction kernel satisfies a certain growth condition, then weak solutions are unique. We present a series of numerical experiments in which we investigate the accuracy of our second-order scheme, demonstrate shock formation in the nonlocal pair-interaction model, and examine how the regularity of the solution depends on the choice of flux function.
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published
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59 (3)
Pages / Article No.
1167 - 1194
Publisher
SIAM
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Subject
hyperbolic conservation laws; nonlocal model; higher-order numerical methods; increased regularity
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03851 - Mishra, Siddhartha / Mishra, Siddhartha
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