AI 中文总结
本文针对双曲守恒律提出基于导数的有限体积Hermite WENO格式，通过在空间重构中排除目标单元导数信息、时间演化采用相同重构多项式来增强鲁棒性，实现统一模板，数值结果验证了该格式的高精度、效率、高分辨率和鲁棒性。
AI 中文摘要
在本文中，我们提出了一种用于双曲守恒律的基于导数的有限体积Hermite WENO（HWENO）格式，其中解及其一阶导数在时间上演化并用于空间重构。求解双曲守恒律的关键挑战是数值解中可能出现的间断。面对间断时，导数可能变得过大，这可能损害HWENO格式的鲁棒性。在最初的HWENO格式中，分别采用不同的模板集来重构控制方程和导数方程，旨在减少导数的影响同时保持高阶精度。然而，这种方法不仅大幅增加计算成本，还引入了相当大的算法复杂性。为克服这些限制，我们在空间重构中排除目标单元导数的信息，同时在时间演化中采用相同的重构多项式来限制导数。该策略增强了传统HWENO格式的鲁棒性，并在基于导数的HWENO框架内允许统一模板。此外，所提出的格式支持任意总和为一的正线性权重并保持紧凑模板。数值结果证明了所提出的HWENO格式的高阶精度、效率、高分辨率和鲁棒性。
英文摘要
In this paper, we propose a derivative-based finite volume Hermite WENO (HWENO) scheme for hyperbolic conservation laws, where both the solution and its first-order derivatives are evolved in time and utilized in spatial reconstructions. The key challenge for solving hyperbolic conservation laws is the possible emergence of discontinuities in the numerical solutions. When facing discontinuities, the derivatives can become excessively large, which may compromise the robustness of HWENO schemes. In the first HWENO scheme, different sets of stencils were adopted for reconstructing the governing equation and the derivative equation, respectively, aiming to reduce the influence of the derivatives while preserving high-order accuracy. However, this approach not only substantially increases computational cost but also introduces considerable algorithmic complexity. To overcome these limitations, we exclude the information of the target cell's derivatives from spatial reconstructions, while employing the same reconstructed polynomial during temporal evolution to limit the derivatives. This strategy enhances the robustness of traditional HWENO schemes and allows unified stencils within the derivative-based HWENO framework. Furthermore, the proposed scheme supports arbitrary positive linear weights that sum to one and maintains a compact stencil. Numerical results demonstrate the high-order accuracy, efficiency, high resolution, and robustness of the proposed HWENO scheme.
Comments31 pages, 10 figures
