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Discontinuous Galerkin reduced basis empirical quadrature procedure for model reduction of parametrized nonlinear conservation laws

  • Masayuki YanoEmail author
Article
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Abstract

We present a model reduction formulation for parametrized nonlinear partial differential equations (PDEs) associated with steady hyperbolic and convection-dominated conservation laws. Our formulation builds on three ingredients: a discontinuous Galerkin (DG) method which provides stability for conservation laws, reduced basis (RB) spaces which provide low-dimensional approximations of the parametric solution manifold, and the empirical quadrature procedure (EQP) which provides hyperreduction of the Galerkin-projection-based reduced model. The hyperreduced system inherits the stability of the DG discretization: (i) energy stability for linear hyperbolic systems, (ii) symmetry and non-negativity for steady linear diffusion systems, and hence (iii) energy stability for linear convection-diffusion systems. In addition, the framework provides (a) a direct quantitative control of the solution error induced by the hyperreduction, (b) efficient and simple hyperreduction posed as a 1 minimization problem, and (c) systematic identification of the reduced bases and the empirical quadrature rule by a greedy algorithm. We demonstrate the formulation for parametrized aerodynamics problems governed by the compressible Euler and Navier-Stokes equations.

Keywords

Parametrized nonlinear PDEs Conservation laws Model reduction Hyperreduction Empirical quadrature Discontinuous Galerkin method 

Mathematics Subject Classification (2010)

65N15 65N30 35Q35 76G25 

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Notes

Acknowledgments

We would like to thank Prof. Anthony Patera (MIT) for many fruitful discussions and the anonymous reviewers for their helpful feedback. We acknowledge the computational resources provided by Compute Canada/SciNet.

Funding information

This study was financially supported by the Natural Sciences and Engineering Research Council of Canada.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.University of TorontoInstitute for Aerospace StudiesTorontoCanada

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