A Kernel-Based Explicit Unconditionally Stable Scheme for Hamilton-Jacobi Equations on Nonuniform Meshes
A Kernel-Based Explicit Unconditionally Stable Scheme for Hamilton-Jacobi Equations on Nonuniform Meshes
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DOI：
10.1016/j.jcp.2020.109543
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发表时间：
2020-02
期刊：
影响因子：
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通讯作者：
A. Christlieb;William A. Sands;Hyoseon Yang
中科院分区：
文献类型：
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作者：
A. Christlieb;William A. Sands;Hyoseon Yang
2010年，作者开发了一类无条件稳定的H-J方程的高阶数值格式，但采用显式格式的形式。本文对这些方案进行了扩展，使其在捕获尖锐梯度时更有效，特别是在非均匀网格上。特别地，我们修改了以前开发的方案中的加权本质非振荡（WENO）方法，通过结合指数基和适应以前开发的用于控制振荡的非线性滤波器。所提方案的主要优点是其有效性和简单性，因为它们可以很容易地在高维非均匀网格上实现。我们对一系列例子进行了数值实验，包括具有线性、非线性、凸和非凸哈密顿量的H-J方程。为了证明所提出方案的灵活性，我们还包括在非平凡几何上定义的测试问题。
In [11], 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.