We present an extension of the summation-by-parts (SBP) framework to tensor-product spectral-element operators in collapsed coordinates. The proposed approach enables the construction of provably stable discretizations of arbitrary order which combine the geometric flexibility of unstructured triangular and tetrahedral meshes with the efficiency of sum-factorization algorithms. Specifically, a methodology is developed for constructing triangular and tetrahedral spectral-element operators of any order which possess the SBP property (i.e. satisfying a discrete analogue of integration by parts) as well as a tensor-product decomposition. Such operators are then employed within the context of discontinuous spectral-element methods based on nodal expansions collocated at the tensor-product quadrature nodes as well as modal expansions employing Proriol-Koornwinder-Dubiner polynomials, the latter approach resolving the time step limitation associated with the singularity of the collapsed coordinate transformation. Energy-stable formulations for curvilinear meshes are obtained using a skew-symmetric splitting of the metric terms, and a weight-adjusted approximation is used to efficiently invert the curvilinear modal mass matrix. The proposed schemes are compared to those using non-tensorial multidimensional SBP operators, and are found to offer comparable accuracy to such schemes in the context of smooth linear advection problems on curved meshes, but at a reduced computational cost for higher polynomial degrees.

翻译：我们提出了将求和积分（SBP）框架推广至退化坐标系下的张量积谱元算子。该方法能够构造任意阶的稳定离散格式，既保留非结构化三角形与四面体网格的几何灵活性，又具备求和因子分解算法的高效性。具体而言，我们发展了任意阶三角形与四面体谱元算子的构造方法，使其同时满足SBP性质（即离散积分分部积分公式的类比）和张量积分解结构。随后，这些算子被应用于两种间断谱元法框架：基于张量积分节点配置的节点展开格式，以及采用Proriol-Koornwinder-Dubiner多项式的模态展开格式——后者通过解耦退化坐标变换的奇异性突破了时间步长限制。通过度量项的反对称分裂获得曲线网格的能量稳定格式，并采用权重调整近似高效求逆曲线模态质量矩阵。与使用非张量多维SBP算子的方案相比，本方法在处理曲线网格上的光滑线性对流问题时具有相当的计算精度，且在高阶多项式情况下显著降低了计算成本。