This work presents a review of high-order hybridisable discontinuous Galerkin (HDG) methods in the context of compressible flows. Moreover, an original unified framework for the derivation of Riemann solvers in hybridised formulations is proposed. This framework includes, for the first time in an HDG context, the HLL and HLLEM Riemann solvers as well as the traditional Lax-Friedrichs and Roe solvers. HLL-type Riemann solvers demonstrate their superiority with respect to Roe in supersonic cases due to their positivity preserving properties. In addition, HLLEM specifically outstands in the approximation of boundary layers because of its shear preservation, which confers it an increased accuracy with respect to HLL and Lax-Friedrichs. A comprehensive set of relevant numerical benchmarks of viscous and inviscid compressible flows is presented. The test cases are used to evaluate the competitiveness of the resulting high-order HDG scheme with the aforementioned Riemann solvers and equipped with a shock treatment technique based on artificial viscosity.

中文翻译：

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可压缩流的可混合不连续伽辽金公式

这项工作回顾了可压缩流背景下的高阶可混合不连续伽辽金 (HDG) 方法。此外，还提出了用于在混合公式中推导黎曼求解器的原始统一框架。该框架首次在 HDG 上下文中包括 HLL 和 HLLEM 黎曼求解器以及传统的 Lax-Friedrichs 和 Roe 求解器。HLL 类型的黎曼求解器证明了它们在超音速情况下相对于 Roe 的优越性，因为它们具有保持正性的特性。此外，HLLEM 在边界层近似方面特别出色，因为它具有剪切保持性，这使其相对于 HLL 和 Lax-Friedrichs 具有更高的准确性。提出了一套综合的粘性和无粘性可压缩流动的相关数值基准。