Second-Order Accurate TVD Numerical Methods for Nonlocal Nonlinear Conservation Laws

We present a second-order accurate numerical method for class of nonlocal nonlinear conservation laws called the "nonlocal pair-interaction model" which was recently introduced by Du, Huang, and LeFloch. Our uses reconstruction-based schemes local in conjunction with appropriate integration. show that resulting is total variation diminishing (TVD) converges towards weak solution. In fact, contrast to laws, our unique entropy solution provided interaction kernel satisfies certain growth condition near zero. Furthermore, as horizon parameter approaches zero we recover well-known laws. addition, answer several questions from paper LeFloch concerning regularity solutions. particular, prove any discontinuity must be stationary that, if condition, then solutions are unique. series experiments investigate accuracy scheme, demonstrate shock formation model, examine how depends on choice flux function.