From spatial to temporal derivatives: A review of the summation-by-parts framework for the development of schemes with provable properties and questions on what the time-marching community can teach us

January 14, 2022
In this talk, I will review the evolution of the summation-by-parts (SBP) framework. Starting from linear PDEs, I will discuss how this framework has matured from its finite-difference origins into an abstract matrix analysis framework that is nearly discretization agnostic for the discretization of spacial derivatives. I will next summarize work by several groups on the link between implicit SBP operators for temporal derivatives and Runge-Kutta (RK) methods and pose a number of questions for discussion. The compelling features of the SBP framework are that it enables the analysis and modification of the actual algorithms implemented in practice (e.g., it accounts for variational crimes such as inexact integration) and leads to the construction of schemes with provable properties (e.g., stability and conservation). I will then move to nonlinear conservation laws where at the continuous level stability can be proven via entropy-stability analysis and demonstrate how these same ideas (and stability proofs) can be constructed leveraging the SBP framework and Tadmor’s two-point flux functions for both space and time. I will then cover alternatives to using implicit SBP operators for temporal derivatives in the form of RK relaxation schemes and (time permitting) a brief review of new work by Yamaleev and Upperman on constructing positivity preserving schemes. I will finish with the nonlinear analogue of the questions I presented in the linear section.
Supplementary Materials