We present a meshless finite difference method for multivariate scalar conservation laws that generates positive schemes satisfying a local maximum principle on irregular nodes and relies on artificial viscosity for shock capturing. Coupling two different numerical differentiation formulas and adaptive selection of the sets of influence allows to meet a local CFL condition without any a priori time step restriction. Artificial viscosity term is chosen in an adaptive way by applying it only in the vicinity of the sharp features of the solution identified by an algorithm for fault detection on scattered data. Numerical tests demonstrate a robust performance of the method on irregular nodes and advantages of adaptive artificial viscosity. The accuracy of the obtained solutions is comparable to that for standard monotone methods available only on Cartesian grids.

翻译：本文提出一种用于多元标量守恒律的无网格有限差分方法，该方法能在不规则节点上生成满足局部最大值原理的正性格式，并依赖人工粘性进行激波捕捉。通过耦合两种不同的数值微分公式及影响域的自适应选择，可在无需先验时间步长限制的情况下满足局部CFL条件。人工粘性项采用自适应方式施加，仅作用于通过散乱数据故障检测算法识别的解尖锐特征附近区域。数值实验表明该方法在不规则节点上具有鲁棒性能，且自适应人工粘性展现出显著优势。所得解的精度可与仅适用于笛卡尔网格的标准单调方法相媲美。