Proper orthogonal decomposition with SUPG-stabilized isogeometric analysis for reduced order modelling of unsteady convection-dominated convection-diffusion-reaction problems

Abstract

We consider reduced order modelling of unsteady convection-dominated convection-diffusion-reaction problems with proper orthogonal decomposition (POD) in combination with isogeometric analysis. Isogeometric analysis has potential advantages in exact geometry representations, efficient mesh generation, different (h, p, and k) refinements and smooth Bspline/NURBS basis functions. In order to compensate the oscillations caused by the convection-dominated effect, the streamline-upwind Petrov-Galerkin (SUPG) stabilization method is used both in generation of snapshots and POD-Galerkin method. Based on the recent novel and promising discretization method-Isogeometric analysis, we propose a new fully discrete SUPG-stabilized scheme, the associated numerical error features three components due to spatial discretization by isogeometric analysis with SUPG stabilization, time discretization with the Crank-Nicolson scheme, and modes truncation by POD. We show a priori error estimates of the fully discrete scheme and give suitable stabilization parameters numerically. A variety of two and three-dimensional benchmark tests and numerical examples are provided to show the effectiveness, accuracy, and efficiency of the reduced order modelling methods by virtue of potential advantages of isogeometric analysis.