Non-linear PetrovGalerkin methods for reduced order modelling of the NavierStokes equations using a mixed finite element pair
A new nonlinear PetrovGalerkin approach has been developed for proper orthogonal decomposition (POD) reduced order modelling (ROM) of the NavierStokes equations. The new method is based on the use of the cosine rule between the advection direction in Cartesian spacetime and the direction of the gradient of the solution. A finite element pair, P1DGP2 P 1 DG P 2 , which has good balance preserving properties is used here, consisting of a mix of discontinuous (for velocity components) and continuous (for pressure) basis functions. The contribution of the present paper lies in applying this new non-linear PetrovGalerkin method to the reduced order NavierStokes equations, and thus improving the stability of ROM results without tuning parameters. The results of numerical tests are presented for a wind driven 2D gyre and the flow past a cylinder, which are simulated using the unstructured mesh finite element CFD model in order to illustrate the numerical performance of the method. The numerical results obtained show that the newly proposed POD PetrovGalerkin method can provide more accurate and stable results than the POD BubnovGalerkin method.
petrovgalerkin navierstokes finite element proper orthogonal decomposition discontinuous-galerkin