Discontinuous Galerkin reduced basis empirical quadrature procedure for model reduction of parametrized nonlinear conservation laws
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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.
KeywordsParametrized nonlinear PDEs Conservation laws Model reduction Hyperreduction Empirical quadrature Discontinuous Galerkin method
Mathematics Subject Classification (2010)65N15 65N30 35Q35 76G25
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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.
This study was financially supported by the Natural Sciences and Engineering Research Council of Canada.
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