We introduce a novel construction procedure for one-dimensional summation-by-parts (SBP) operators. Existing construction procedures for FSBP operators of the form $D = P^{-1} Q$ proceed as follows: Given a boundary operator $B$, the norm matrix $P$ is first determined and then in a second step the complementary matrix $Q$ is calculated to finally get the FSBP operator $D$. In contrast, the approach proposed here determines the norm and complementary matrices, $P$ and $Q$, simultaneously by solving an optimization problem. The proposed construction procedure applies to classical SBP operators based on polynomial approximation and the broader class of function space SBP (FSBP) operators. According to our experiments, the presented approach yields a numerically stable construction procedure and FSBP operators with higher accuracy for diagonal norm difference operators at the boundaries than the traditional approach. Through numerical simulations, we highlight the advantages of our proposed technique.

翻译：本文提出了一种一维求和分部（SBP）算子的新型构造方法。现有形如 $D = P^{-1} Q$ 的FSBP算子构造过程如下：给定边界算子 $B$，首先确定范数矩阵 $P$，然后在第二步中计算互补矩阵 $Q$，最终得到FSBP算子 $D$。与此相反，本文提出的方法通过求解优化问题同时确定范数矩阵和互补矩阵 $P$ 与 $Q$。该构造方法适用于基于多项式逼近的经典SBP算子以及更广泛的函数空间SBP（FSBP）算子。实验表明，与传统方法相比，本文方法在边界处对角范数差分算子的构造过程数值稳定且能获得更高精度的FSBP算子。通过数值模拟，我们展示了所提技术的优势。