Robust DG Schemes on Unstructured Triangular Meshes: Oscillation Elimination and Bound Preservation via Optimal Convex Decomposition

Abstract

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.

Figures (22)

• Figure 1: Illustration and notations of the element $K\in \mathcal{T}_h$ and its adjacent elements.
• Figure 2: Visualization of the critical polynomial $p_*(x,y)$ defined in \ref{['eq:1773']} on an example triangular cell $K$ with vertices $(0,0)$, $(4,0)$, and $(3,2)$.
• Figure 3: Comparison of BP CFL numbers $\mathcal{C}^{\tt DCW}_{K,k}$, $\mathcal{C}^{\tt ZXS}_{K,k}$, and $\mathcal{C}^{\tt CS}_{K,k}$ on general triangular cells $K$, $k=1,2$. Without loss of generality, assume $1 = l^{(1)}_K\ge l^{(2)}_K\ge l^{(3)}_K$ and $l^{(1)}_K\le l^{(2)}_K + l^{(3)}_K$. Please note that $\mathcal{C}^{\tt CS}_{K,1} = \mathcal{C}^{\tt CS}_{K,2} = 1/6$ in this case.
• Figure 4: (\ref{['Ex:LinSmooth']}) The meshes $\mathcal{T}_{h_0}$ and $\mathcal{T}_{h_1}$.
• Figure 5: (\ref{['Ex:LinDis']}) Contours of the numerical solutions obtained by using the $\mathbb{P}^k$-based OEDG schemes, $k \in \{1,2,3,4\}$.

Theorems & Definitions (42)

• Remark 1: Compactness
• Remark 2: Simplicity and Efficiency
• Definition 3.1
• Example 3.1: 2D Compressible Euler System
• Definition 3.2
• Theorem 3.3
• proof
• Theorem 3.4: Rotational Invariance
• proof
• Definition 4.1: Feasible convex decomposition on a triangular cell $K$