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

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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 ℓ1 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.

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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

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

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Authors and Affiliations

  1. University of Toronto, Institute for Aerospace Studies, 4925 Duffein Street, Toronto, ON, M3H 5T6, Canada

    Masayuki Yano

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  1. Masayuki Yano
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Correspondence to Masayuki Yano.

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Communicated by: Anthony Nouy

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Yano, M. Discontinuous Galerkin reduced basis empirical quadrature procedure for model reduction of parametrized nonlinear conservation laws. Adv Comput Math 45, 2287–2320 (2019). https://doi.org/10.1007/s10444-019-09710-z

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