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Mathematics Colloquium

FSU and San Diego State

Title: Summation by parts, a discrete integral calculus, and how we are making discontinuous Galerkin spectral element methods that work.
Date: Wednesday, October 3, 2018 (Note special day.)
Place and Time: Room 101, Love Building, 3:35-4:25 pm
Refreshments: Room 204, Love Building, 3:00 pm

Abstract. Discontinuous Galerkin spectral element methods (DGSEMs) have advanced to the point where open source solvers are available to compute solutions of industrial grade problems. It is also known that DGSEMs can go unstable, especially during under-resolved simulations like those that occur in turbulence computations. We have developed a discrete integral calculus framework for the analysis of the stability of DGSEMs that starts with summation-by-parts -- the discrete analog of integration by parts. The framework allows us to identify when approximations are stable or not, and has enabled us to derive provably stable methods for one-, two- or three-dimensional geometries, unstructured grids, curved elements, moving or static meshes, and linear and nonlinear systems of equations. In this talk, I will describe summation-by-parts, and show how it can be used to generate a discrete integral calculus framework. I will then give examples of the use of the framework for the analysis of the stability of DGSEM approximations to conservation laws.