A Split-Form, Stable, Hybrid Continuous/Discontinuous Galerkin Spectral Element Method for Wave Propagation

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If you would like to join the seminar and are not currently affiliated with ANU, please contact Kenneth Duru at kenneth.duru@anu.edu.au.


Spectral element methods, which have desired features of low dissipation and dispersion errors and spectral or even exponential convergence rates, have been used for many years to compute wave propagation problems. The discontinuous Galerkin (DG) spectral element method is robust and adds dissipation at element boundaries through a numerical flux. The continuous Galerkin (CG) version naturally manages inter-element connectivity without a numerical flux and without dissipation. A hybrid method of the two is being used in the electromagnetics community under the name of the CGDGTD (CG/DG Time Domain) method that leverages the advantages of each, such as reduced dissipation from the CG side, and parallel efficiency and discontinuous media capabilities from the DG side.

This talk will present a provably stable hybrid CG/DG method for hyperbolic systems that uses a split form of the equations and two-point fluxes to ensure stability for unstructured quadrilateral/hexahedral meshes with curved elements in three space dimensions. The approximation is conservative and constant state preserving on such meshes. 

The properties are established through the use of a discrete integral calculus that follows from the summation-by-parts property of Gauss quadratures. Part of the talk will review the framework, which allows one to match continuous analysis on the PDE with discrete analysis on the approximation.


David A. Kopriva is a Professor Emeritus in the Department of Mathematics at the Florida State University, and an Adjunct Professor in the Computational Science Research Center at San Diego State University. He did his Ph.D in the Applied Mathematics Program at the University of Arizona and his postdoc with Yousuff Hussaini at ICASE, the Institute for Computer Applications in Science and Engineering, where he developed the idea to use multidomain spectral methods. He has taught at MIT, FSU, IIT and SDSU, and was a Program Manager in Applied Mathematics and Computational Mathematics at the NSF. His research has focussed on the development of spectral element methods for hyperbolic and advection dominated problems with applications in computational fluid dynamics, aerodynamics, electromagnetics and aeroacoustics. With his collaborators in Europe and Scandinavia, his interests have most recently been on the development of robust, provably stable methods for such problems through the use of split forms and discontinuous Galerkin methods that have the Summation-By-Parts property. These methods have been applied to the compressible Navier-Stokes equations, the shallow water equations, the MHD equations and two-phase flow problems. He also has interests in grid generation, motivated by his needs over the years for meshes suitable for spectral element methods.