Second-order accurate TVD numerical methods for nonlocal nonlinear conservation laws
Ulrik Skre Fjordholm, Adrian Montgomery Ruf
备注：
26 pages, 19 figures, 9 tables; The article is published in SIAM Journal on Numerical Analysis. The final publication is available at https://doi.org/10.1137/20M1360979
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.
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。
