Embedded WENO methods utilize all adjacent smooth substencils to construct a desirable interpolation. Conventional WENO schemes under-use this possibility close to large gradients or discontinuities. We develop a general approach for constructing embedded versions of existing WENO schemes. Embedded methods based on the WENO schemes of Jiang and Shu (J. Comput. Phys. 126 (1996)) and on the WENO-Z scheme of Borges et al. (J. Comput. Phys. 227 (2008)) are explicitly constructed. Several possible choices are presented that result in either better spectral properties or a higher order of convergence for sufficiently smooth solutions. However, these improvements carry over to discontinuous solutions. The embedded methods are demonstrated to be indeed improvements over their standard counterparts by several numerical examples. All the embedded methods presented have no added computational effort compared to their standard counterparts.
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。
An Adaptive Characteristic-wise Reconstruction WENOZ scheme for Gas Dynamic Euler Equations
Institute of Applied Physics and Computational Mathematics（应用物理与计算力学研究所）；State Key Laboratory of High Temperature Gas Dynamics, Institute of Mechanics, Chinese Academy of Sciences（中国科学院力学研究所高温气体动力学国家重点实验室）
King Abdullah University of Science & Technology（阿卜杜拉国王科技大学）；Mathematical Institute, University of Oxford（牛津大学数学研究所）；Department of Mathematics, Simon Fraser University（西蒙菲莎大学数学系）
