用于可压缩纳维-斯托克斯方程的高效且鲁棒的五阶梯度重构HWENO格式
An efficient and robust fifth-order HWENO scheme with gradient reconstruction for compressible Navier--Stokes equations
AI总结：
本文提出一种带梯度重构的五阶HWENO格式，用于可压缩纳维-斯托克斯方程，该格式兼具高效性与鲁棒性，可稳定求解含强梯度的流动，数值结果验证了其精度与性能。
AI中文摘要：
本文针对可压缩纳维-斯托克斯方程，提出了一种高效且鲁棒的五阶有限体积厄米加权基本无振荡（HWENO）格式，该格式带有梯度重构。其核心思路是：对每个耗散变量，通过取两个HWENO界面迹线的算术平均值来构造弱导数矩，随后通过标准标量非线性HWENO重构公式按分量处理这些矩，仅输入矩与对流重构所用的矩不同，以此提供黏性通量所需的梯度。该重构仅使用次数不超过4次的候选多项式即可达到五阶精度，避免了直接求导会导致的阶数降低问题，且引入的额外计算成本可忽略不计。为确保鲁棒性，尤其是在涉及强激波或低密度的极端测试案例中，我们对守恒态应用保正限制器，同时保留用于黏性梯度的非线性HWENO重构。这种组合对于强梯度流动中的稳定计算至关重要。守恒重构和梯度重构共享相同的紧致模板和候选多项式结构，可实现单一实现例程，而不会引入额外的算法复杂性。数值结果证实，所提格式可达到五阶精度并具有高分辨率，具备有竞争力的计算效率，且能鲁棒地求解挑战性的有限雷诺数纳维-斯托克斯流动，此时保正限制器处于主动作用状态。
英文摘要：
In this paper, we propose an efficient and robust fifth-order finite-volume Hermite weighted essentially non-oscillatory (HWENO) scheme with gradient reconstruction for the compressible Navier--Stokes equations. The key idea is to construct weak-derivative moments for each dissipative variable by taking the arithmetic average of two HWENO interface traces, which are then processed componentwise through the standard scalar nonlinear HWENO reconstruction formula, with only the input moments differing from those used in the convective reconstruction, thereby supplying the gradients needed for the viscous fluxes. This reconstruction achieves fifth-order accuracy using only candidate polynomials of degree at most four, avoids the order reduction that would result from direct differentiation, and introduces negligible additional computational cost. To ensure robustness, especially in extreme test cases involving strong shocks or low densities, we apply a positivity-preserving limiter to the conservative states while retaining the nonlinear HWENO reconstruction for the viscous gradients. This combination is essential for stable computations in flows with strong gradients. Both the conservative and gradient reconstructions share the same compact stencils and candidate-polynomial structure, enabling a single implementation routine without introducing additional algorithmic complexity. Numerical results confirm that the proposed scheme delivers fifth-order accuracy and high resolution, offers competitive computational efficiency, and robustly resolves challenging finite-Reynolds-number Navier--Stokes flows in which the positivity-preserving limiter is actively engaged.
