# Some Notes on Summation by Parts Time Integration Methods

## TL;DR

This paper investigates the stability and properties of summation by parts (SBP) time integration methods, revealing their connections to Runge-Kutta schemes and clarifying the conditions under which they possess desirable stability features.

## Contribution

It provides new proofs and counterexamples to clarify the assumptions, relations, and stability properties of SBP schemes and their connection to Runge-Kutta methods.

## Key findings

- A necessary technical assumption for SBP schemes is identified.
- Not all Runge-Kutta schemes with SBP stability are derived from SBP schemes.
- Classical collocation methods on Radau and Lobatto nodes are SBP schemes.

## Abstract

Some properties of numerical time integration methods using summation by parts operators and simultaneous approximation terms are studied. These schemes can be interpreted as implicit Runge-Kutta methods with desirable stability properties such as $A$-, $B$-, $L$-, and algebraic stability. Here, insights into the necessity of certain assumptions, relations to known Runge-Kutta methods, and stability properties are provided by new proofs and counterexamples. In particular, it is proved that a) a technical assumption is necessary since it is not fulfilled by every SBP scheme, b) not every Runge-Kutta scheme having the stability properties of SBP schemes is given in this way, c) the classical collocation methods on Radau and Lobatto nodes are SBP schemes, and d) nearly no SBP scheme is strong stability preserving.