In this study, we provide a novel third-order weighted essentially non-oscillatory (WENO) method to solve Hamilton-Jacobi equations. The key idea is to incorporate exponential polynomials to construct numerical fluxes and smoothness indicators. First, the new smoothness indicators are designed by using the finite difference operator annihilating exponential polynomials such that singular regions can be distinguished from smooth regions more efficiently. Moreover, to construct numerical flux, we employ an interpolation method based on exponential polynomials which yields improved results around steep gradients. The proposed scheme retains the optimal order of accuracy (i.e., three) in smooth areas, even near the critical points. To illustrate the ability of the new scheme, some numerical results are provided along with comparisons with other WENO schemes.

在这项研究中，我们提供了一种新颖的三阶加权基本非振荡（WENO）方法来求解Hamilton-Jacobi方程。关键思想是合并指数多项式以构造数字通量和平滑度指标。首先，通过使用有限差分算子消除指数多项式来设计新的平滑度指标，以便可以更有效地将奇异区域与平滑区域区分开。此外，为了构造数值通量，我们采用了基于指数多项式的插值方法，该方法在陡峭的梯度附近产生改进的结果。所提出的方案即使在临界点附近也能在光滑区域中保持最佳的精度顺序（即三个）。为了说明新方案的功能，