Non-linear Petrov-Galerkin methods for reduced order hyperbolic equations and discontinuous finite element methods

Authors:
F. Fang;C. C. Pain;I. M. Navon;A. H. Elsheikh;J. Du;D. Xiao
Affiliations:
Applied Modelling and Computation Group, Department of Earth Science and Engineering, Imperial College London, Prince Consort Road, London SW7 2BP, UK;Applied Modelling and Computation Group, Department of Earth Science and Engineering, Imperial College London, Prince Consort Road, London SW7 2BP, UK;Department of Scientific Computing, Florida State University, Tallahassee, FL 32306-4120, USA;Applied Modelling and Computation Group, Department of Earth Science and Engineering, Imperial College London, Prince Consort Road, London SW7 2BP, UK;Applied Modelling and Computation Group, Department of Earth Science and Engineering, Imperial College London, Prince Consort Road, London SW7 2BP, UK and The Institute of Atmospheric Physics, The ...;Applied Modelling and Computation Group, Department of Earth Science and Engineering, Imperial College London, Prince Consort Road, London SW7 2BP, UK
Venue:
Journal of Computational Physics
Year:
2013

Citing 4
Cited 2

A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations

Journal of Computational Physics
Optimal control of flow with discontinuities

Journal of Computational Physics
Enablers for robust POD models

Journal of Computational Physics
A trajectory piecewise-linear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices

IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems

The GNAT method for nonlinear model reduction: Effective implementation and application to computational fluid dynamics and turbulent flows

Journal of Computational Physics
Non-linear model reduction for the Navier-Stokes equations using residual DEIM method

Journal of Computational Physics

Quantified Score

Hi-index	31.46

Visualization

Abstract

A new Petrov-Galerkin approach for dealing with sharp or abrupt field changes in discontinuous Galerkin (DG) reduced order modelling (ROM) is outlined in this paper. This method presents a natural and easy way to introduce a diffusion term into ROM without tuning/optimising and provides appropriate modelling and stablisation for the numerical solution of high order nonlinear PDEs. The approach is based on the use of the cosine rule between the advection direction in Cartesian space-time and the direction of the gradient of the solution. The stabilization of the proper orthogonal decomposition (POD) model using the new Petrov-Galerkin approach is demonstrated in 1D and 2D advection and 1D shock wave cases. Error estimation is carried out for evaluating the accuracy of the Petrov-Galerkin POD model. Numerical results show the new nonlinear Petrov-Galerkin method is a promising approach for stablisation of reduced order modelling.