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Model order reduction for parametrized nonlinear hyperbolic problems as an application to uncertainty quantification. (English) Zbl 1404.65112

Summary: In this work, we present a model order reduction (MOR) technique for hyperbolic conservation laws with applications in uncertainty quantification (UQ). The problem consists of a parametrized time dependent hyperbolic system of equations, where the parameters affect the initial conditions and the fluxes in a non- linear way. The procedure utilized to reduce the order is a combination of a Greedy algorithm in the parameter space, a proper orthogonal decomposition (POD) in time and empirical interpolation method (EIM) to deal with non-linearities (Drohmann, 2012). We provide under some hypothesis an error bound for the reduced solution with respect to the high order one. The algorithm shows small errors and savings of the computational time up to 90% in the UQ simulations, which are performed to validate the algorithm.

MSC:

65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65J15 Numerical solutions to equations with nonlinear operators
76L05 Shock waves and blast waves in fluid mechanics
35L65 Hyperbolic conservation laws
35R60 PDEs with randomness, stochastic partial differential equations

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