Discontinuous Galerkin (DG) schemes on unstructured meshes offer the advantages of compactness and the ability to handle complex computational domains. However, their robustness and reliability in solving hyperbolic conservation laws depend on two critical abilities: suppressing spurious oscillations and preserving intrinsic bounds or constraints. This paper introduces two significant advancements in enhancing the robustness and efficiency of DG methods on unstructured meshes for general hyperbolic conservation laws, while maintaining their accuracy and compactness. First, we investigate the oscillation-eliminating (OE) DG methods on unstructured meshes. These methods not only maintain key features such as conservation, scale invariance, and evolution invariance but also achieve rotation invariance through a novel rotation-invariant OE (RIOE) procedure. Second, we propose, for the first time, the optimal convex decomposition for designing efficient bound-preserving (BP) DG schemes on unstructured meshes. Finding the optimal convex decomposition that maximizes the BP CFL number is an important yet challenging problem.While this challenge was addressed for rectangular meshes, it remains an open problem for triangular meshes. This paper successfully constructs the optimal convex decomposition for the widely used $P^1$ and $P^2$ spaces on triangular cells, significantly improving the efficiency of BP DG methods.The maximum BP CFL numbers are increased by 100%--200% for $P^1$ and 280.38%--350% for $P^2$, compared to classic decomposition. Furthermore, our RIOE procedure and optimal decomposition technique can be integrated into existing DG codes with little and localized modifications. These techniques require only edge-neighboring cell information, thereby retaining the compactness and high parallel efficiency of original DG methods.

翻译：非结构网格上的间断伽辽金（DG）格式具有紧致性和处理复杂计算域的优势。然而，其在求解双曲守恒律时的鲁棒性与可靠性取决于两个关键能力：抑制伪振荡和保持固有界或约束。本文针对一般双曲守恒律，在保持精度与紧致性的前提下，提出了两项提升非结构网格上DG方法鲁棒性与效率的重要进展。首先，我们研究了非结构网格上的振荡消除（OE）DG方法。这些方法不仅保持了守恒性、尺度不变性和演化不变性等关键特性，还通过一种新颖的旋转不变OE（RIOE）过程实现了旋转不变性。其次，我们首次提出了用于设计非结构网格上高效保界（BP）DG格式的最优凸分解。寻找能够最大化BP CFL数的最优凸分解是一个重要但具有挑战性的问题。虽然该挑战在矩形网格上已得到解决，但在三角形网格上仍是一个开放性问题。本文成功地为三角形单元上广泛使用的$P^1$和$P^2$空间构造了最优凸分解，显著提升了BP DG方法的效率。与经典分解相比，最大BP CFL数在$P^1$情况下提高了100%--200%，在$P^2$情况下提高了280.38%--350%。此外，我们的RIOE过程和最优分解技术只需极少且局部的修改即可集成到现有DG代码中。这些技术仅需相邻单元边信息，从而保留了原始DG方法的紧致性和高并行效率。