论文标题
绝对收敛的固定点快速扫描WENO方法，用于双曲线保护法的稳态
Absolutely convergent fixed-point fast sweeping WENO methods for steady state of hyperbolic conservation laws
论文作者
论文摘要
在文献中开发了固定点迭代的清扫方法，以有效地解决汉密尔顿 - 雅各比方程和双曲线保护法的稳态解决方案。与其他快速扫描方案相似，此类方法的关键组成部分是高斯式迭代和交替的扫描策略，以实现快速收敛速度。此外，定点迭代扫描方法的良好特性包括它们具有明确的形式，并且不涉及非线性局部系统的逆操作，并且可以使用任何单调数值通量和高阶近似值将其应用于一般双曲线方程。在[L。 Wu，Y.-T。张，S。Zhang和C.-W。 Shu，Commun。计算。 Phys。，20（2016）]，设计了第五阶固定点扫描WENO方案，并且显示该方案的收敛速度比总变化减小（TVD）Runge-Kutta方法快得多，通过稳定的高阶方案，具有前向EULER时制定的高阶方案。一个开放的问题是，对于某些基准数值示例，固定点扫描WENO方案的迭代残基悬挂在截断误差级别上，而不是安置为机器零。这个问题使得很难确定迭代的收敛标准，并在将方法应用于复杂问题上。为了解决这个问题，在本文中，我们应用了[J.朱和C.-W。 Shu，J。Comput。 Phys。，375（2018）]到第五阶固定点扫描WENO方案，并获得了一种绝对收敛的固定点快速扫描方法，用于稳态双曲线保护定律，即，快速扫描迭代的残留物会收敛到所有基准问题的机器零 /圆形偏移错误。
Fixed-point iterative sweeping methods were developed in the literature to efficiently solve steady state solutions of Hamilton-Jacobi equations and hyperbolic conservation laws. Similar as other fast sweeping schemes, the key components of this class of methods are the Gauss-Seidel iterations and alternating sweeping strategy to achieve fast convergence rate. Furthermore, good properties of fixed-point iterative sweeping methods include that they have explicit forms and do not involve inverse operation of nonlinear local systems, and they can be applied to general hyperbolic equations using any monotone numerical fluxes and high order approximations easily. In [L. Wu, Y.-T. Zhang, S. Zhang and C.-W. Shu, Commun. Comput. Phys., 20 (2016)], a fifth order fixed-point sweeping WENO scheme was designed and it was shown that the scheme converges much faster than the total variation diminishing (TVD) Runge-Kutta approach by stability improvement of high order schemes with a forward Euler time-marching. An open problem is that for some benchmark numerical examples, the iteration residue of the fixed-point sweeping WENO scheme hangs at a truncation error level instead of settling down to machine zero. This issue makes it difficult to determine the convergence criterion for the iteration and challenging to apply the method to complex problems. To solve this issue, in this paper we apply the multi-resolution WENO scheme developed in [J. Zhu and C.-W. Shu, J. Comput. Phys., 375 (2018)] to the fifth order fixed-point sweeping WENO scheme and obtain an absolutely convergent fixed-point fast sweeping method for steady state of hyperbolic conservation laws, i.e., the residue of the fast sweeping iterations converges to machine zero / round off errors for all benchmark problems tested.
