Balsara's genuinely multidimensional Riemann solvers, which possess a simple closed-form and high efficiency in the low-order cases, realize their multidimensional effect successfully by introducing the vertex-based framework. However, traditional higher-order (third-order and above) reconstruction methods built for the structured grids become ambiguous when applied directly into these solvers. Nowadays, there is little research on the higher-order reconstruction methods for such Riemann solvers besides Balsara's work by adopting the ADER-WENO schemes. Based on Balsara's work, we conduct research on the multidimensional higher-order reconstruction procedure in this study. This paper investigates two third-order reconstruction methods with the WENO approach for the genuinely two-dimensional HLLE Riemann solver. The critical idea of the reconstruction methods is to utilize the solution or its derivative to construct the spatial two-dimensional interpolation polynomials with the third-order accuracy. The first reconstruction method denoted by TWENO constructs the polynomials by the Taylor expansion, while the second one denoted by HWENO adopts the Hermite interpolation polynomials. By combining the polynomials with the WENO limiter, these schemes can provide required reconstructed values at the midpoints and corner-points of the cell interfaces for the framework and avoid oscillations near discontinuities. A discontinuity-detector technology is adopted to improve the computational efficiency of these reconstruction methods in simulating complex flows. Several numerical test cases are conducted and show that these third-order methods achieve their design accuracy and capture the shock waves with no overshoots or oscillations. The multidimensional nature of the solutions is kept well by the TWENO/HWENO methods, and the computational time is reduced by about 30% with the discontinuity-detector technology. Moreover, it is found that the genuinely two-dimensional Riemann solver fits the third-order reconstruction methods naturally due to its unique application of a Simpson rule.

Balsara真正的多维黎曼求解器，在低阶情况下具有简单的封闭形式和高效率，通过引入基于顶点的框架成功地实现了它们的多维效果。然而，当直接应用于这些求解器时，为结构化网格构建的传统高阶（三阶及以上）重建方法变得模棱两可。目前，除了Balsara采用ADER-WENO方案的工作外，对此类黎曼求解器的高阶重构方法的研究很少。基于 Balsara 的工作，我们对本研究中的多维高阶重建过程进行了研究。本文研究了用于真正二维 HLLE 黎曼求解器的两种三阶重建方法，其中使用 WENO 方法。重构方法的关键思想是利用解或其导数构造具有三阶精度的空间二维插值多项式。TWENO表示的第一种重构方法通过泰勒展开式构造多项式，而HWENO表示的第二种重构方法采用Hermite插值多项式。通过将多项式与 WENO 限制器相结合，这些方案可以在框架的单元界面的中点和角点处提供所需的重建值，并避免在不连续点附近发生振荡。采用不连续检测器技术来提高这些重建方法在模拟复杂流中的计算效率。进行了几个数值测试案例，并表明这些三阶方法达到了设计精度并捕获了没有超调或振荡的冲击波。TWENO/HWENO 方法很好地保持了解的多维性质，并且使用不连续检测器技术将计算时间减少了约 30%。此外，发现真正的二维黎曼求解器由于其对辛普森规则的独特应用而自然地适合三阶重建方法。