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

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.