We derive and study a Lax--Friedrichs type finite volume method for a large class of nonlocal continuity equations in multiple dimensions. We prove that the method converges weakly to the measure-valued solution, and converges strongly if the initial data is of bounded variation. Several numerical examples for the kinetic Kuramoto equation are provided, demonstrating that the method works well both for regular and singular data.
A fifth-order absolutely convergent fixed-point fast sweeping hybrid alternative WENO scheme for steady state of hyperbolic conservation laws
Huang Huai University（黄淮大学）；Nanjing University of Aeronautics and Astronautics（南京航空航天大学）；Indiana University South Bend（南本德印第安纳大学）；University of Notre Dame（圣母大学）
Absolutely convergent fixed-point fast sweeping WENO methods for steady state of hyperbolic conservation laws
文献中发展了不动点迭代扫描方法来有效地求解Hamilton-Jacobi方程和双曲守恒律的稳态解。与其他快速扫描方案类似，这类方法的关键组成部分是Gauss-Seidel迭代和交替扫描策略，以实现快速收敛速度。此外，不动点迭代扫描方法具有显式形式和不涉及非线性局部系统的逆运算等优点，并且可以方便地应用于一般双曲型方程的任意单调数值通量和高阶近似。在[L.吴玉- T. Zhang，S. Zhang和C.- W. Shu，Commun. Comput.物理、[20（2016）]，设计了一种五阶不动点扫描WENO格式，通过对高阶格式的稳定性改进和前向Euler时间推进，证明了该格式比全变差递减（TVD）Runge-Kutta方法收敛更快。一个公开的问题是，对于一些基准数值例子，迭代残留的不动点扫描WENO计划挂在截断误差的水平，而不是安定下来的机器零。这个问题使得它很难确定迭代的收敛标准，并具有挑战性的方法应用到复杂的问题。为了解决这个问题，本文采用了[J. Zhu和C.-]中提出的多分辨率WENO格式。W. Shu，J. Comput.物理、375（2018）]推广到五阶定点扫描WENO格式，得到了双曲守恒律定常解的一个绝对收敛的不动点快速扫描方法，即，对于所有测试的基准问题，快速扫描迭代的残差收敛到机器零/舍入误差。
A fully convergent fixed-point fast sweeping method with the WENO-JS local solver for steady state of hyperbolic conservation laws
School of Mathematics and Statistics, Huang Huai University（河南工程学院数学与统计学院）；College of Science and Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA), MIIT, Nanjing University of Aeronautics and Astronautics（南京航空航天大学理学院及工信部航空器数学建模与高性能计算重点实验室）；Department of Applied and Computational Mathematics and Statistics, University of Notre Dame（圣母大学应用与计算数学和统计系）
The finite precision computation and the nonconvergence of difference scheme
State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics (LASG), Institute of Atmospheric Physics, Chinese Academy of Sciences（中国科学院大气物理研究所数值模拟大气科学和地球流体力学国家重点实验室）
