Title: Through the lens of summation by parts: A matrix analysis framework for developing numerical methods with provable properties
Biography:
David C. Del Rey Fernández is an Associate Professor in the Department of Applied Mathematics at the University of Waterloo. He is the Pratt & Whitney Canada, Research Chair in Industrial Artificial Intelligence, an Associate Director of the Waterloo Institute for Sustainable Aeronautics, and a Research Cluster Lead (Modelling) for the Future Cities Institute. He is a recipient of the Industrial Mathematics Society Early Career Award (2024) as well as the Faculty of Mathematics Golden Jubilee Research Excellence Award (2025).
Prof. Del Rey Fernández is the Graduate Officer of the Computational Mathematics Program, on the Advisory Committee for Women in Mathematics at the University of Waterloo as well as a member of the Canvassing Committee for Association for Women in Mathematics. He obtained his PhD from the University of Toronto Institute for Aerospace Studies (2015) and was a postdoctoral fellow and then research scientist (2016-2021) at NASA Langley Research Center through the National Institute for Aerospace.
Prof. Del Rey Fernández works broadly on the development of numerical methods for partial differential equations (PDEs) with mathematical guarantees, with projects in the aerospace sector with industry partners such as Pratt & Whitney Canada, Bombardier, and Ansys, collaborators in government labs (e.g., NRC, NASA, INRIA, Sandia National Laboratory, Argonne National laboratory, etc.), and various academic partners. In particular, his work has focused on the summation-by-parts (SBP) framework for solving linear and nonlinear PDEs. The SBP concept consists of constructing SBP operators that are discretely mimetic of integration by parts and extending this property to the full discretization. Doing so enables a one-to-one correspondence between continuous proofs (e.g., stability and conservation) and discrete proofs. Extending the SBP concept has been a key priority of his work and has culminated in a very general theory of the SBP concept that enables the analysis and design of a broad set of existing and novel discretization approaches. This framework has been extended to provably-stable (entropy-stable) discretizations for nonlinear PDEs and his work in recent years has focused on developing these schemes (a key component of the NASA CFD Vision 2030 Study) enabling both h, p, and r adaptation (important in developing efficient schemes for multiscale problems) as well as extending entropy-stability approaches to multidimensional SBP operators (important in simplifying the meshing of complex geometry and hence applicability of the methods to real-world problems); the resulting schemes have demonstrated their superior performance on real-world problems and are a significant step in productionizing high-order methods for practical simulation. Recent research directions include quantum algorithms for PDEs in partnership with NRC as well as developing machine learning algorithms with mathematical guarantees.
Abstract:
Simulation tools, and the numerical methods that power them, are essential in modern science and engineering permeating the design and analysis of the simplest to the most sophisticated engineering systems. Simulation tools are used throughout product development cycles, from the ideation phase all the way through final product design and certification. Despite tremendous effort and many successes, the development of efficient numerical methods with mathematical guarantees remains a difficult task, particularly in the context of nonlinear PDEs on complex, potentially moving and deforming, geometries, and efficient deployment on modern computer hardware. Nearly 50 years ago, summation-by-parts (SBP) operators were created by Kreiss and Scherer (1974) in order to bring to finite-difference methods the ability to systematically prove stability common in many finite-element approaches. Since then, and continuously drawing inspiration from the FEM community, the SBP concept has matured into an abstract matrix based approach for analyzing a wide range of existing schemes, the development of novel schemes, and the design-order modification of schemes such that the result is provably stable. The matrix point of view gives incredible flexibility in designing and developing schemes. In recent years, these ideas have been extended to the notion of entropy-stability and have resulted in provably-stable schemes for a variety of nonlinear PDEs important in fluid mechanics (e.g., compressible Euler and Navier-Stokes equations) and resulted in high-order schemes that are sufficiently robust to apply to industry level problems. In this talk, I will provide a high-level introduction to the SBP framework and the current state-of-the art. I will then cover a number of recent developments in my group leveraging the SBP idea including shock-tracking schemes, methods that preserve asymptotic stability, and SBP adjacent schemes such as the residual correction approach (among other topics). I will also touch on the opportunities and difficulties in developing numerical methods on quantum computers that retain provable properties. Time permitting, I will touch on other efforts in my group including on the fly reduced-order modeling, and machine learning accelerated simulation workflows.