We introduce a robust first order accurate meshfree method to numerically solve time-dependent nonlinear conservation laws. The main contribution of this work is the meshfree construction of first order consistent summation by parts differentiations. We describe how to efficiently construct such operators on a point cloud. We then study the performance of such differentiations, and then combine these operators with a numerical flux-based formulation to approximate the solution of nonlinear conservation laws, with focus on the advection equation and the compressible Euler equations. We observe numerically that, while the resulting mesh-free differentiation operators are only $O(h^\frac{1}{2})$ accurate in the $L^2$ norm, they achieve $O(h)$ rates of convergence when applied to the numerical solution of PDEs.

翻译：本文提出了一种鲁棒的一阶精度无网格方法，用于数值求解时变非线性守恒律。本工作的主要贡献在于构建了具有一阶相容性的分部求和微分算子。我们描述了如何在点云上高效构造此类算子，并研究了其性能表现。随后，将这些算子与基于数值通量的离散格式相结合，以近似求解非线性守恒律，重点关注对流方程和可压缩欧拉方程。数值实验表明，虽然所构造的无网格微分算子在$L^2$范数下仅具有$O(h^\frac{1}{2})$精度，但在应用于偏微分方程数值解时能够达到$O(h)$的收敛阶。