Some reduced finite difference schemes based on a proper orthogonal decomposition technique for parabolic equations

In this paper, some reduced finite difference schemes based on a proper orthogonal decomposition (POD) technique for parabolic equations are derived. Also the error estimates between the POD approximate solutions of the reduced finite difference schemes and the exact solutions for parabolic equations are established. It is shown by considering the results of two numerical examples that the numerical results are consistent with theoretical conclusions. Moreover, it is also shown that the POD reduced finite difference schemes are feasible and efficient.

[1]  Yukio Tamura,et al.  Dynamic wind pressures acting on a tall building model — proper orthogonal decomposition , 1997 .

[2]  Zhendong Luo,et al.  Finite difference scheme based on proper orthogonal decomposition for the nonstationary Navier-Stokes equations , 2007 .

[3]  S. Ravindran.  A reduced-order approach for optimal control of fluids using proper orthogonal decomposition , 2000 .

[4]  Arthur Veldman,et al.  Proper orthogonal decomposition and low-dimensional models for driven cavity flows , 1998 .

[5]  I. Jolliffe.  Principal Component Analysis , 2002 .

[6]  James R. DeBonis,et al.  Application of Proper Orthogonal Decomposition to a Supersonic Axisymmetric Jet , 2003 .

[7]  George Em Karniadakis,et al.  A low-dimensional model for simulating three-dimensional cylinder flow , 2002, Journal of Fluid Mechanics.

[8]  David J. Lucia,et al.  Reduced Order Modeling for a One-Dimensional Nozzle Flow with Moving Shocks , 2001 .

[9]  H. Tran,et al.  Proper Orthogonal Decomposition for Flow Calculations and Optimal Control in a Horizontal CVD Reactor , 2002 .

[10]  K. Kunisch,et al.  Control of the Burgers Equation by a Reduced-Order Approach Using Proper Orthogonal Decomposition , 1999 .

[11]  Zhendong Luo,et al.  Reduced-Order Modeling of the Upper Tropical Pacific Ocean Model using Proper Orthogonal Decomposition , 2006, Comput. Math. Appl..

[12]  Frank M. Selten,et al.  Baroclinic empirical orthogonal functions as basis functions in an atmospheric model , 1997 .

[13]  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 .

[14]  Ionel Michael Navon,et al.  An Equation-Free Reduced-Order Modeling Approach to Tropical Pacific Simulation , 2009 .

[15]  Parviz Moin,et al.  Characteristic-eddy decomposition of turbulence in a channel , 1989, Journal of Fluid Mechanics.

[16]  Stefan Volkwein,et al.  Galerkin proper orthogonal decomposition methods for parabolic problems , 2001, Numerische Mathematik.

[17]  Stig Larsson,et al.  Partial differential equations with numerical methods , 2003, Texts in applied mathematics.

[18]  P. Holmes,et al.  Turbulence, Coherent Structures, Dynamical Systems and Symmetry , 1996 .

[19]  L. Sirovich,et al.  Low-dimensional description of free-shear-flow coherent structures and their dynamical behaviour , 1994, Journal of Fluid Mechanics.

[20]  Paul G. A. Cizmas,et al.  Proper Orthogonal Decomposition of Turbine Rotor-Stator Interaction , 2003 .

[21]  Ionel M. Navon,et al.  A reduced‐order approach to four‐dimensional variational data assimilation using proper orthogonal decomposition , 2007 .

[22]  Fabio D'Andrea,et al.  Extratropical low-frequency variability as a low-dimensional problem , 2001 .

[23]  Keinosuke Fukunaga,et al.  Introduction to statistical pattern recognition (2nd ed.) , 1990 .

[24]  Ionel M. Navon,et al.  An optimizing reduced order FDS for the tropical Pacific Ocean reduced gravity model , 2007 .

[25]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.