Summation-by-parts (SBP) operators are popular building blocks for systematically developing stable and high-order accurate numerical methods for time-dependent differential equations. The main idea behind existing SBP operators is that the solution is assumed to be well approximated by polynomials up to a certain degree, and the SBP operator should therefore be exact for them. However, polynomials might not provide the best approximation for some problems, and other approximation spaces may be more appropriate. In this paper, a theory for SBP operators based on general function spaces is developed. We demonstrate that most of the established results for polynomial-based SBP operators carry over to this general class of SBP operators. Our findings imply that the concept of SBP operators can be applied to a significantly larger class of methods than currently known. We exemplify the general theory by considering trigonometric, exponential, and radial basis functions.

翻译：和分部分（SBP）算子是系统构建时间依赖微分方程稳定且高精度数值方法的重要基础工具。现有SBP算子的核心思想假设解可通过特定阶数的多项式进行良好逼近，因此SBP算子应对这些多项式具有精确性。然而，对于某些问题，多项式可能并非最佳逼近方式，其他逼近空间可能更为适用。本文发展了基于通用函数空间的SBP算子理论，证明多数基于多项式的SBP算子的经典结论可推广至此类广义SBP算子。研究结果表明，SBP算子的应用范围可显著扩展至远超当前已知的方法体系。我们通过三角函数、指数函数和径向基函数实例阐明了该通用理论。