Essentially non-oscillatory (ENO) and weighted ENO (WENO) schemes were designed for solving hyperbolic and convection–diffusion equations with possibly discontinuous solutions or solutions with sharp gradient regions. The main idea of ENO and WENO schemes is actually an approximation procedure, aimed at achieving arbitrarily high-order accuracy in smooth regions and resolving shocks or other discontinuities sharply and in an essentially non-oscillatory fashion. Both finite volume and finite difference schemes have been designed using the ENO or WENO procedure, and these schemes are very popular in applications, most noticeably in computational fluid dynamics but also in other areas of computational physics and engineering. Since the main idea of the ENO and WENO schemes is an approximation procedure not directly related to partial differential equations (PDEs), ENO and WENO schemes also have non-PDE applications. In this paper we will survey the basic ideas behind ENO and WENO schemes, discuss their properties, and present examples of their applications to different types of PDEs as well as to non-PDE problems.

中文翻译：

---

基本非振荡和加权基本非振荡方案

本质上非振荡 (ENO) 和加权 ENO (WENO) 方案设计用于求解具有可能不连续解或具有陡梯度区域的解的双曲线和对流扩散方程。ENO 和 WENO 方案的主要思想实际上是一个近似过程，旨在在平滑区域实现任意高阶精度，并以一种基本上非振荡的方式急剧地解决冲击或其他不连续性。有限体积和有限差分方案都是使用 ENO 或 WENO 程序设计的，这些方案在应用中非常流行，尤其是在计算流体动力学以及计算物理和工程的其他领域。由于 ENO 和 WENO 方案的主要思想是与偏微分方程 (PDE) 没有直接关系的近似过程，因此 ENO 和 WENO 方案也有非 PDE 应用。在本文中，我们将调查 ENO 和 WENO 方案背后的基本思想，讨论它们的性质，并展示它们在不同类型的 PDE 以及非 PDE 问题中的应用示例。