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. 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 from Du, Huang, and LeFloch 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.
翻译:我们为一类非本地非线性养护法提供了一种第二阶准确的数字方法,称为“非本地对口互动模式”,这是杜、黄和勒弗洛奇最近推出的。我们的数字方法对本地养护法采用了第二阶级精确的重建计划,同时进行了适当的数字整合。我们从文件中了解到,由此产生的方法是完全变异的减少(TVD),并趋于一个薄弱的解决办法。事实上,与地方养护法不同,我们的基于二级重建的方法与独特的银河式解决办法相交,条件是非本地互动内核满足接近零的某种增长条件。此外,随着我们方法中的非本地地平线参数接近零,我们恢复了当地养护法的众所周知的二级方法。此外,我们还回答了杜、黄和莱弗洛奇关于解决办法的规律性的若干问题。特别是,我们证明,薄弱解决办法中的任何不连续性都必须是固定的,如果互动内核满足某种增长条件,那么薄弱的解决办法是独一无二的。我们提出了一系列的数值实验,我们用来调查当地选择方法的精确性模式,我们如何在正常的版本中决定着电压式办法的形成。