论文标题
一般功能空间的逐件求和运算符
Summation-by-parts operators for general function spaces
论文作者
论文摘要
逐个组合（SBP）运算符是用于系统地开发稳定且高阶准确的数值方法的流行构件，用于时间相关的微分方程。现有SBP操作员背后的主要思想是，该解决方案被假定为一定程度的多项式近似，因此SBP操作员应为其确切。但是，多项式可能无法为某些问题提供最佳近似值，而其他近似空间可能更合适。在本文中，开发了基于一般功能空间的SBP操作员的理论。我们证明，基于多项式的SBP运算符的大多数既定结果都将其转移到了SBP运营商的一般类别中。我们的发现表明，SBP运营商的概念可以应用于比目前已知的更大的方法。我们通过考虑三角学，指数和径向基础功能来体现一般理论。
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.
