Scalable Evaluation of Hadamard Products with Tensor Product Basis for Entropy-Stable High-Order Methods
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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 𝒪(n^d+1) flops rather than 𝒪(n^2d), 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 𝒪(n^d+1) 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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