论文标题
基于内核的明确稳定方案，用于汉密尔顿 - 雅各比方程，在不均匀网格上
A Kernel-Based Explicit Unconditionally Stable Scheme for Hamilton-Jacobi Equations on Nonuniform Meshes
论文作者
论文摘要
在\ cite {Christlieb2019Kernel}中，作者为汉密尔顿 - 雅各布（H-J）方程开发了一类高阶数值方案，它们是无条件稳定的，但采用了显式方案的形式。本文扩展了此类方案，因此它们在捕获尖锐的梯度方面更有效，尤其是在不均匀的网格上。特别是，我们通过合并指数基础并适应用于控制振荡的先前开发的非线性过滤器，在先前开发的方案中修改加权基本非振荡（WENO）方法。所提出的方案的主要优点是它们的有效性和简单性，因为它们可以轻松地在高维非均匀网格上实施。我们对示例集进行数值实验，包括带有线性，非线性，凸和非凸的Hamiltonians的H-J方程。为了证明所提出的方案的灵活性，我们还包括在非平凡几何形状上定义的测试问题。
In \cite{christlieb2019kernel}, the authors developed a class of high-order numerical schemes for the Hamilton-Jacobi (H-J) equations, which are unconditionally stable, yet take the form of an explicit scheme. This paper extends such schemes, so that they are more effective at capturing sharp gradients, especially on nonuniform meshes. In particular, we modify the weighted essentially non-oscillatory (WENO) methodology in the previously developed schemes by incorporating an exponential basis and adapting the previously developed nonlinear filters used to control oscillations. The main advantages of the proposed schemes are their effectiveness and simplicity, since they can be easily implemented on higher-dimensional nonuniform meshes. We perform numerical experiments on a collection of examples, including H-J equations with linear, nonlinear, convex and non-convex Hamiltonians. To demonstrate the flexibility of the proposed schemes, we also include test problems defined on non-trivial geometry.
