AI 中文总结
研究针对三角形网格上双曲守恒律，开发含ADER时间离散化的紧凑简单HWENO格式。核心方法是利用Lax-Wendroff转换及格林-高斯定理获取导数单元平均值，避免额外方程求解，每次时间步仅一次HWENO重构。主要贡献是更高效、误差小、模板紧凑，时空精度高且间断处无振荡。
AI 中文摘要
本文为三角形网格上的双曲守恒律开发了一种紧凑高阶的HWENO格式，采用ADER（利用导数的任意高阶）时间离散化，是对结构化网格工作的扩展。利用Lax-Wendroff过程将时间导数转换为空间导数，通过格林-高斯定理由时间精确解获取解导数的单元平均值。与现有非结构化网格上的Runge-Kutta HWENO方法相比，新方法无需求解额外方程，每次时间步仅进行一次HWENO重构，更高效且数值误差小、计算成本低。与同精度的现有ADER-WENO方法相比，新方法模板更紧凑。数值例子表明新方法在时空上对光滑解能达到高阶精度，在间断附近保持无振荡。
英文摘要
A compact and high order HWENO scheme using ADER (Arbitrary high order using DERivatives) time discretization is developed for hyperbolic conservation laws on the triangular mesh, which is the extension of the work on the structured mesh (Luo et. al. (2024) \cite{luo2023}). The Lax-Wendroff procedure is employed to convert time derivatives to spatial derivatives. Thanks to this, the cell averages of the derivatives of the solution can be obtained by the time accurate solution as Gaussian points along the cell interfaces through the Green-Gauss theorem instead of by the evolution solution directly in the conventional HWENO methods. Comparing with the existing Runge-Kutta HWENO (RK-HWENO) method on the unstructured mesh (Zhao et. al. (2025) \cite{zhao2025}), the new method has the following advantages. Firstly, the RK-HWENO method must solve the additional equations for reconstructions and time advancing, which is avoided for the new method. Secondly, the HWENO reconstruction in the new method is performed once per time step and is different from the RK-HWENO method, in which the reconstruction is performed several times every time step. Because of these advantages the new method is more efficient than the RK-HWENO method with smaller numerical errors and less computational costs. Besides, comparing with the existing ADER-WENO methods \cite{dumbser20071,dumbser20072} under the same order of accuracy, the stencil of the new method is more compact since the both the function and its first derivative values are used in the reconstruction of the HWENO schemes. Numerical examples demonstrate that the new method can achieve the high order for smooth solutions both in space and time, keep non-oscillatory near discontinuities.
Comments30 Pages, 13 figures
