Topic: 85th CFD Seminar: A Brief Introduction to Summation-By-Parts Methods and Their Various Flavors
Date: Tuesday, April 11, 2017
Time: 11:00am-noon (EST)
Room: NIA, Rm137
Speaker: David Del Rey Fernández, Postdoctoral Fellow, NIA
Abstract: The concept of matrix difference operators having the summation-by-parts (SBP) property has its origin in the finite-difference (FD) community with the goal of mimicking finite-element energy analysis techniques for proving linear stability. The essential feature of these operators is that they are equipped with a high-order approximation to integration by parts. When combined with appropriate procedures for inter-element coupling and imposition of boundary conditions, the resulting SBP framework allows for a one-to-one correspondence between discrete and continuous stability proofs and in this way naturally guides the construction of robust algorithms.
In recent years, there has been a veritable explosion in the SBP concept. The SBP framework has been applied to nodal discontinuous and continuous Galerkin methods, the flux-reconstruction method, and has been shown to have a subcell finite-volume interpretation. The SBP concept has been extended to non-tensor nodal distributions thereby introducing the ability to construct SBP schemes on unstructured tetrahedral meshes. Nonlinearly robust schemes have been constructed by enforcing discrete entropy stability. SBP schemes on nonconforming meshes that remain conservative and stable have been developed. Dual-consistent schemes have been developed that lead to superconvergent functional estimates, etc. In summary, the SBP concept enables the construction of flexible and robust numerical methods that have advantageous properties within a rigorous mathematical framework.
In this talk I will give a brief introduction to the SBP concept, starting with the tensor-product variety and its various flavors. I will then show how these ideas extend to non-tensor nodal distributions and will finish with some remarks on nonlinear stability.
Bio: Dr. David Del Rey Fernández obtained his PhD in aerospace engineering from the University of Toronto Institute for Aerospace Studies (UTIAS) in 2015. He remained at UTIAS until December 2016 as a Postdoctoral Fellow and lecturer. Currently he is a Postdoctoral Fellow at NIA and NASA LaRC in the Computational AeroScience Branch. His research interests revolve around the development of flexible and robust high-order numerical methods for the solution of partial differential equations. Currently he is working on developing architecture-aware nonlinearly stable summation-by-parts methods applicable to structured and unstructured meshes that retain their conservation and nonlinear stability properties under various adaptive strategies such as AMR.