Abstract High-order methods gain more and more attention in computational fluid dynamics. Among these, spectral element methods and discontinuous Galerkin methods introduce element-wise approximations by means of a polynomial basis. This leads to a small number of operators consuming a large portion of the runtime of CFD applications. The present paper addresses tensor-product bases which are among the most frequent in these applications. Various implementations are developed and performance tests conducted for the interpolation operator, the Helmholtz operator, and the fast diagonalization operator. For each, up to 50% of the peak performance is attained, beating matrix-matrix multiplication for every polynomial degree relevant for simulations. This extremely high efficiency of the method developed is then demonstrated on a combustion problem with 1.72 · 109 mesh points.

中文翻译：

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用于 CFD 构建块算子的高效高阶谱元离散化

摘要 高阶方法在计算流体动力学中越来越受到重视。其中，谱元素方法和不连续伽辽金方法通过多项式基础引入了元素近似。这导致少数操作员消耗大部分 CFD 应用程序运行时间。本论文讨论了这些应用中最常见的张量积基。为插值算子、亥姆霍兹算子和快速对角化算子开发了各种实现并进行了性能测试。对于每个，达到峰值性能的 50%，击败与模拟相关的每个多项式次数的矩阵-矩阵乘法。