时间依赖非线性守恒律的鲁棒一阶无网格方法
A robust first order meshfree method for time-dependent nonlinear conservation laws
AI总结：
提出一种鲁棒的一阶无网格方法，通过构造点云上的一阶相容求和分部微分算子并结合数值通量公式，求解非线性守恒律，数值实验表明该方法在PDE求解中达到一阶收敛。
摘要：
我们提出了一种鲁棒的一阶精确无网格方法，用于数值求解时间依赖的非线性守恒律。本工作的主要贡献在于无网格地构造一阶相容的求和分部（summation by parts）微分算子。我们描述了如何在点云上高效地构造此类算子。随后，我们研究了此类微分的性能，并将这些算子与基于数值通量的公式相结合，以近似非线性守恒律的解，重点关注平流方程和可压缩欧拉方程。我们通过数值实验观察到，虽然所得的无网格微分算子在$L^2$范数下仅具有$O(h^{\ rac{1}{2}})$的精度，但在应用于偏微分方程的数值解时，它们达到了$O(h)$的收敛阶。
英文摘要：
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.
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