标题： 基于张量积基的Hadamard乘积的可扩展评估用于熵稳定高阶方法 显示英文标题

标题： Scalable Evaluation of Hadamard Products with Tensor Product Basis for Entropy-Stable High-Order Methods

摘要： 一种用于在张量积基下评估Hadamard乘积的求和-因子化形式在此工作中被推导出来。 所提出的算法允许Hadamard乘积在$\mathcal{O}\left(n^{d+1}\right)$次浮点运算中计算，而不是$\mathcal{O}\left(n^{2d}\right)$次，其中$d$是问题的维度。 通过这一改进，需要在一般模态情况下进行密集Hadamard乘积的熵守恒且稳定的格式，与模态不连续Galerkin (DG)格式相比，在计算上变得具有竞争力。 我们通过基于非线性稳定通量重构方案的自研偏微分方程求解器PHiLiP，数值地展示了求和-因子化Hadamard乘积的应用。 我们证明了熵守恒流动求解器在曲线坐标系下的三维可压缩流中可以达到$\mathcal{O}\left(n^{d+1}\right)$的扩展性，并与模态DG和过积分DG格式进行了计算成本比较。 显示英文摘要

摘要： A sum-factorization form for the evaluation of Hadamard products with a tensor product basis is derived in this work. The proposed algorithm allows for Hadamard products to be computed in $\mathcal{O}\left(n^{d+1}\right)$ flops rather than $\mathcal{O}\left(n^{2d}\right)$, where $d$ is the dimension of the problem. With this improvement, entropy conserving and stable schemes, that require a dense Hadamard product in the general modal case, become computationally competitive with the modal discontinuous Galerkin (DG) scheme. We numerically demonstrate the application of the sum-factorized Hadamard product in our in-house partial differential equation solver PHiLiP based on the Nonlinearly Stable Flux Reconstruction scheme. We demonstrate that the entropy conserving flow solver scales at $\mathcal{O}\left(n^{d+1}\right)$ for three-dimensional compressible flow in curvilinear coordinates, along with a computational cost comparison with the modal DG and over-integrated DG schemes.