In this paper, we investigate numerical approximations of the scalar conservation law with the Caputo derivative, which introduces the memory effect. We construct the first order and the second order explicit upwind schemes for such equations, which are shown to be conditionally $\ell^1$ contracting and TVD. However, the Caputo derivative leads to the modified CFL-type stability condition, $ (Δt)^α = O(Δx)$, where $α\in (0,1]$ is the fractional exponent in the derivative. When $α$ small, such strong constraint makes the numerical implementation extremely impractical. We have then proposed the implicit upwind scheme to overcome this issue, which is proved to be unconditionally $\ell^1$ contracting and TVD. Various numerical tests are presented to validate the properties of the methods and provide more numerical evidence in interpreting the memory effect in conservation laws.
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 Structure- and Pressure-Positivity-Preserving Semi-implicit IMEX Finite Volume Scheme for Ideal MHD at All Acoustic Mach and Alfvén Mach Numbers with Generic Equation of State
University of Cambridge（剑桥大学）；Université Savoie Mont Blanc（萨瓦勃朗西大学）；University of Ferrara（费拉拉大学）
