Some Notes on Summation by Parts Time Integration Methods
Hendrik Ranocha
TL;DR
The paper analyzes SBP-SAT time discretizations for ODEs and their interpretation as implicit Runge-Kutta methods, focusing on stability properties such as $A$-, $L$-, and $B$-stability under a key eigenvalue assumption. It shows that some SBP operators violate the eigenvalue condition and that not every stability-rich RK scheme arises from SBP-SAT, while also proving that classical Radau and Lobatto collocation schemes are captured by SBP-SAT. The SSP analysis reveals strong limitations: otherwise, SBP-SAT with diagonal $M$ generally cannot be SSP under standard accuracy or endpoint conditions, though a rare counterexample exists with a specially crafted operator. Together, these results delineate the capabilities and limitations of SBP-SAT time integration for stiff and dissipative problems, guiding design of stable, high-order integrators.
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.