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We present a second-order accurate numerical method for a class of nonlocalnonlinear conservation laws called the "nonlocal pair-interaction model" whichwas recently introduced by Du, Huang, and LeFloch. Our numerical method usessecond-order accurate reconstruction-based schemes for local conservation lawsin conjunction with appropriate numerical integration. We show that theresulting method is total variation diminishing (TVD) and converges towards aweak solution. In fact, in contrast to local conservation laws, oursecond-order reconstruction-based method converges towards the unique entropysolution provided that the nonlocal interaction kernel satisfies a certaingrowth condition near zero. Furthermore, as the nonlocal horizon parameter inour method approaches zero we recover a well-known second-order method forlocal conservation laws. In addition, we answer several questions from thepaper from Du, Huang, and LeFloch concerning regularity of solutions. Inparticular, we prove that any discontinuity present in a weak solution must bestationary and that, if the interaction kernel satisfies a certain growthcondition, then weak solutions are unique. We present a series of numericalexperiments in which we investigate the accuracy of our second-order scheme,demonstrate shock formation in the nonlocal pair-interaction model, and examinehow the regularity of the solution depends on the choice of flux function.

Second-order accurate TVD numerical methods for nonlocal nonlinear conservation laws