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

• Published in: Advances in computational mathematics Vol. 45; no. 5-6; pp. 2287 - 2320
• Main Author: Yano, Masayuki
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2019; Springer Nature B.V
• Subjects: Compressibility; Computational fluid dynamics; Computational mathematics; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Conservation laws; Convection; Dimensional stability; Galerkin method; Greedy algorithms; Hyperbolic systems; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematical models; Mathematics; Mathematics and Statistics; Model reduction; Model reduction of parametrized Systems; Nonlinear differential equations; Nonlinear equations; Parameterization; Partial differential equations; Quadratures; Visualization; Conservation laws; Discontinuous Galerkin method; 65N15; Hyperreduction; Empirical quadrature; 35Q35; Model reduction; 76G25; 65N30; Parametrized nonlinear PDEs
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-019-09710-z

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 ℓ 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.