AI 中文总结
研究为一般函数空间构建优化的分部求和算子，利用改进的广义高斯求积算法，引入优化策略，经数值测试，新算子在精度上远超标准算子，凸显算子优化对准确高效离散化的关键作用。
AI 中文摘要
我们为一般函数空间构造了优化的分部求和（SBP）算子，这些算子在开、闭和半开节点分布上具有可证明的最小自由度。这些算子依赖于广义高斯求积规则，我们提出了一种灵活、高效且可证明收敛的改进算法。在有自由参数的情况下，我们进一步引入了两种算子优化策略。通过一些包含大梯度或无界梯度的数值例子进行测试，新算子在解的精度相对于自由度方面比标准多项式算子高出几个数量级，且显著优于等距节点分布的函数空间SBP算子。最后表明算子优化过程对于实现准确高效的离散化至关重要，标准SBP构造过程可能导致零空间不一致和条件不佳的算子。
英文摘要
We construct optimized summation-by-parts (SBP) operators for general function spaces with provably minimal degrees of freedom on open, closed, and half-open nodal distributions. These operators rely on generalized Gaussian quadrature rules, for which we present an improved algorithm that is flexible, efficient, and provably convergent. In cases where free parameters are available, we further introduce two operator optimization strategies. We test our operators on a handful of numerical examples that contain large or unbounded gradients, in which some a priori knowledge of the solution has been assumed to select an appropriate basis. The novel operators are found to outperform standard polynomial operators by several orders of magnitude in solution accuracy relative to degrees of freedom. Furthermore, our novel operators significantly outperform function-space SBP operators with equispaced nodal distributions, which require significantly more nodes for the same operator basis. Finally, we demonstrate that the operator optimization procedures are critical to achieving accurate and efficient discretizations, as the standard SBP construction procedure can lead to nullspace-inconsistent and poorly-conditioned operators.
CommentsReproducibility repository: https://github.com/alexbercik/Paper-GaussFSBP
