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Seminar | Mathematics and Computer Science Division

The Summation-By-Parts Framework: An Abstract Matrix-Analysis Approach to the Development of Discrete Schemes with Provable Properties

LANS Seminar

Nonlinear partial differential equations (PDEs) are of fundamental importance in basic science and engineering. However, their numerical solution remains a highly active area of research. For example, in aerospace, the accurate simulation of the compressible Navier-Stokes (NS) equations is critical. While significant progress has been made in the last number of decades, state-of-the-art Reynolds-Averaged NS solvers fail to be predictive for important portions of the flight envelope (e.g., low- speed/high-lift, stall, buffet, flutter, and shocks). Simultaneously, current and future compute hardware promises unprecedented power. This power comes via highly parallel architectures with complex memory hierarchies, the efficient use of which demands dense compute kernels, communication hiding, etc. High-order methods are a natural choice for such systems, however, their use for practical problems has been stymied by stability issues. 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. 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 emphasize that its utility is not only for developing novel cutting-edge algorithms but also for the analysis and design-order modification of productionized codes. 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.

 

Bio: Dr. David C. Del Rey Fernandez received his Ph.D. in 2015 from the University of Toronto Institute for Aerospace Studies and completed a postdoc at NASA Langley Research Center where, until recently, he worked as a research scientist. He is currently an assistant professor in the Department of Applied Mathematics at the University of Waterloo.