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：本文提出了求和性质框架在塌缩坐标下张量积谱元算子的扩展。所提出的方法能够构建任意阶数可证明稳定的离散格式，将非结构化三角形和四面体网格的几何灵活性与求和因子化算法的高效性相结合。具体而言，我们开发了一种构建任意阶三角形和四面体谱元算子的方法学，这些算子同时具备SBP性质（即满足分部积分离散模拟）和张量积分解特性。此类算子随后被应用于基于节点展开（配置在张量积求积节点）和模态展开（采用Proriol-Koornwinder-Dubiner多项式）的间断谱元法中，后一种方法解决了由塌缩坐标变换奇异性引起的时间步长限制问题。通过度量项的斜对称分裂，获得了曲线网格的能量稳定格式，并采用权重调整近似法高效求逆曲线模态质量矩阵。将所提方案与使用非张量多维SBP算子的方案进行比较，发现在弯曲网格上的光滑线性平流问题中，两者精度相当，但在较高多项式阶数下本文方案计算成本更低。