We review and extend the list of stability and convergence properties satisfied by Runge-Kutta (RK) methods that are associated with Summation-By-Parts (SBP) operators, herein called RK-SBP methods. The analysis covers classical, generalized as well as upwind SBP operators. Previous work on the topic has relied predominantly on energy estimates. In contrast, we derive all results using a purely algebraic approach that rests on the well-established theory of RK methods.

The purpose of this paper is to provide a bottom-up overview of stability and convergence results for linear and non-linear problems that relate to general RK-SBP methods. To this end, we focus on the RK viewpoint, since this perspective so far is largely unexplored. This approach allows us to derive all results as simple consequences of the properties of SBP methods combined with well-known results from RK theory. In this way, new proofs of known results such as A-, L- and B-stability are given. Additionally, we establish previously unreported results such as strong S-stability, dissipative stability and stiff accuracy of certain RK-SBP methods. Further, it is shown that a subset of methods are B-convergent for strictly contractive non-linear problems and convergent for non-linear problems that are both contractive and dissipative.

我们回顾并扩展了由Runge-Kutta（RK）方法满足的稳定性和收敛性的列表，这些方法与按部分求和（SBP）运算符相关联，在此称为RK-SBP方法。该分析涵盖了经典，广义以及逆风SBP运算符。以前有关该主题的工作主要依靠能量估计。相反，我们使用完全基于RK方法的理论的纯代数方法得出所有结果。

本文的目的是提供与常规RK-SBP方法有关的线性和非线性问题的稳定性和收敛性结果的自底向上概述。为此，我们将重点放在RK观点上，因为到目前为止，这一观点在很大程度上尚未得到探索。这种方法使我们能够将所有结果作为SBP方法属性的简单结果与RK理论的知名结果相结合得出。这样，给出了已知结果的新证明，例如A，L和B稳定性。此外，我们建立了以前未报告的结果，例如某些RK-SBP方法的强S稳定性，耗散稳定性和刚性精度。此外，还表明，对于严格收缩的非线性问题，方法的子集是B收敛的；对于收缩性和耗散性的非线性问题，方法的子集是B收敛的。