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.

翻译：本文推导了采用张量积基评估Hadamard积的和因式分解形式。所提算法使得Hadamard积的计算复杂度从$\mathcal{O}\left(n^{2d}\right)$浮点运算降低至$\mathcal{O}\left(n^{d+1}\right)$，其中$d$为问题维度。通过这一改进，在一般模态情形下需要密集Hadamard积的熵守恒与熵稳定格式在计算效率上变得与模态间断伽辽金(DG)方法具有竞争力。我们基于非线性稳定通量重构框架，在自主开发的偏微分方程求解器PHiLiP中数值验证了和因式分解Hadamard积的应用。结果表明，熵守恒流动求解器在曲线坐标系下求解三维可压缩流动时，其计算复杂度达到$\mathcal{O}\left(n^{d+1}\right)$，同时给出了与模态DG及过积分DG方法的计算成本对比分析。