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arXiv · 2306.11665

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

Alexander Cicchino·Siva Nadarajah
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Abstract

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.

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Alexander Cicchino, Siva Nadarajah. 2023-06-20. Scalable Evaluation of Hadamard Products with Tensor Product Basis for Entropy-Stable High-Order Methods. https://arxiv.org/abs/2306.11665

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