Stabilized reduced order models for the advection-diffusion-reaction equation using operator splitting

Reduced order modeling (ROM) coupled with finite element methods has been used effectively in many disciplines to efficiently solve complex problems. However, for advection-dominated flows numerical simulations often contain spurious, nonphysical oscillations which will also be apparent in the ROM simulations. In this work we consider stabilization methods for ROM for the advection-diffusion-reaction (ADR) equation when it is solved both with and without operator splitting. Specifically we consider the streamline-upwind Petrov-Galerkin (SUPG) stabilization method and the spurious oscillations at layers diminishing (SOLD) stabilization method. We build on these methods by constructing a coherent framework which successfully integrates these model reduction, stabilization, and operator splitting approaches, and we provide numerical examples detailing the application of this framework in the ADR setting. The stabilized ROM results are compared numerically with their corresponding full finite element simulations.

[1]  Max D. Gunzburger,et al.  Centroidal Voronoi Tessellation-Based Reduced-Order Modeling of Complex Systems , 2006, SIAM J. Sci. Comput..

[2]  T. Hughes,et al.  Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations , 1990 .

[3]  Ionel Michael Navon,et al.  Proper orthogonal decomposition approach and error estimation of mixed finite element methods for the tropical Pacific Ocean reduced gravity model , 2007 .

[4]  Volker John,et al.  On spurious oscillations at layers diminishing (SOLD) methods for convection–diffusion equations: Part I – A review , 2007 .

[5]  Stefan Turek,et al.  Flux correction tools for finite elements , 2002 .

[6]  Ramon Codina,et al.  A discontinuity-capturing crosswind-dissipation for the finite element solution of the convection-diffusion equation , 1993 .

[7]  J. Peraire,et al.  Balanced Model Reduction via the Proper Orthogonal Decomposition , 2002 .

[8]  Jay P. Boris,et al.  Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works , 1973 .

[9]  John N. Shadid,et al.  Stability of operator splitting methods for systems with indefinite operators: Advection-diffusion-reaction systems , 2009, J. Comput. Phys..

[10]  L. Franca,et al.  Stabilized Finite Element Methods , 1993 .

[11]  Qiang Du,et al.  Centroidal Voronoi Tessellations: Applications and Algorithms , 1999, SIAM Rev..

[12]  J. Boris,et al.  Flux-Corrected Transport , 1997 .

[13]  S. Zalesak.  Fully multidimensional flux-corrected transport algorithms for fluids , 1979 .

[14]  Traian Iliescu,et al.  SUPG reduced order models for convection-dominated convection–diffusion–reaction equations , 2014 .

[15]  Petr Knobloch,et al.  On the definition of the SUPG parameter. , 2008 .

[16]  M. Gunzburger,et al.  Reduced-order modeling of time-dependent PDEs with multiple parameters in the boundary data , 2007 .

[17]  T. Hughes,et al.  Stabilized finite element methods. I: Application to the advective-diffusive model , 1992 .

[18]  Ahmed K. Noor,et al.  Reduced Basis Technique for Nonlinear Analysis of Structures , 1980 .

[19]  Pavel B. Bochev,et al.  Stability of the SUPG finite element method for transient advection-diffusion problems , 2004 .

[20]  Arif Masud,et al.  Revisiting stabilized finite element methods for the advective–diffusive equation , 2006 .