论文标题
静态汉密尔顿 - 雅各比方程的高阶有限差异HERMITE WENO快速扫描方法
High order finite difference Hermite WENO fast sweeping methods for static Hamilton-Jacobi equations
论文作者
论文摘要
在本文中，我们提出了一种新型的Hermite加权基本上是非振荡（HWENO）快速扫描方法，以有效地解决静态的Hamilton-Jacobi方程。在HWENO重建程序中，提出的方法建立在新的有限差五阶Hweno方案上，该方案涉及一支大型模板和两个小模板。但是，与传统的Hweno框架相比，一个主要的新颖性和差异在于，我们不需要引入和求解任何其他方程来更新未知函数$ ϕ $的衍生物。取而代之的是，我们使用当前的$ ϕ $和$ ϕ $的旧空间导数来更新它们。在本文中还引入了传统的Hweno快速扫描方法，以进行比较，其中引入了$ ϕ $的空间衍生物的其他方程式。新型的Hweno快速扫描方法显示出可以在计算时间和存储中节省大量，从而提高了传统Hweno方案的计算效率。此外，还引入了一种混合策略，以进一步降低计算成本。提供了广泛的数值实验，以验证所提出方法的准确性和效率。
In this paper, we propose a novel Hermite weighted essentially non-oscillatory (HWENO) fast sweeping method to solve the static Hamilton-Jacobi equations efficiently. During the HWENO reconstruction procedure, the proposed method is built upon a new finite difference fifth order HWENO scheme involving one big stencil and two small stencils. However, one major novelty and difference from the traditional HWENO framework lies in the fact that, we do not need to introduce and solve any additional equations to update the derivatives of the unknown function $ϕ$. Instead, we use the current $ϕ$ and the old spatial derivative of $ϕ$ to update them. The traditional HWENO fast sweeping method is also introduced in this paper for comparison, where additional equations governing the spatial derivatives of $ϕ$ are introduced. The novel HWENO fast sweeping methods are shown to yield great savings in both computational time and storage, which improves the computational efficiency of the traditional HWENO scheme. In addition, a hybrid strategy is also introduced to further reduce computational costs. Extensive numerical experiments are provided to validate the accuracy and efficiency of the proposed approaches.
