Reduced-order extrapolation spectral-finite difference scheme based on POD method and error estimation for three-dimensional parabolic equation

Abstract

In this study, a classical spectral-finite difference scheme (SFDS) for the three-dimensional (3D) parabolic equation is reduced by using proper orthogonal decomposition (POD) and singular value decomposition (SVD). First, the 3D parabolic equation is discretized in spatial variables by using spectral collocation method and the discrete scheme is transformed into matrix formulation by tensor product. Second, the classical SFDS is obtained by difference discretization in time-direction. The ensemble of data are comprised with the first few transient solutions of the classical SFDS for the 3D parabolic equation and the POD bases of ensemble of data are generated by using POD technique and SVD. The unknown quantities of the classical SFDS are replaced with the linear combination of POD bases and a reducedorder extrapolation SFDS with lower dimensions and sufficiently high accuracy for the 3D parabolic equation is established. Third, the error estimates between the classical SFDS solutions and the reduced-order extrapolation SFDS solutions and the implementation for solving the reduced-order extrapolation SFDS are provided. Finally, a numerical example shows that the errors of numerical computations are consistent with the theoretical results. Moreover, it is shown that the reduced-order extrapolation SFDS is effective and feasible to find the numerical solutions for the 3D parabolic equation.

Keywords

Singular value decomposition (SVD) / proper orthogonal decomposition (POD) bases / spectral-finite difference scheme (SFDS) / error estimation / parabolic equation

References

[1] Afanasiev K, Hinze M. Adaptive control of a wake flow using proper orthogonal decomposition. Lect Notes Pure Appl Math, 2001, 216: 317-332

[2] Algazi V, Sakrison D. On the optimality of Karhunen-Lo`eve expansion. IEEE Trans Inform Theory, 1969, 15: 319-321

[3] Arian E, Fahl M, Sachs E W. Trust-region proper orthogonal decomposition models by optimization method. In: Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, Nevada, 2002. 2002, 3300-3305

[4] Aubry N, Holmes P, Lumley J L, Stone E. The dynamics of coherent structures in the wall region of a turbulent boundary layer. J Fluid Dynamics, 1988, 192: 115-173

[5] Cao Y H, Zhu J, Luo Z H, Navon I M. Reduced order modeling of the upper tropical Pacific Ocean model using proper orthogonal decomposition. Comput Math Appl, 2006, 52: 1373-1386

[6] Cao Y H, Zhu J, Navon I M, Luo Z D. A reduced order approach to four-dimensional variational data assimilation using proper orthogonal decomposition. Int J Numer Meth Fluids, 2007, 53: 1571-1583

[7] Fox L, Parker I B. Chebyshev Polynomials in Numerical Analysis. Oxford: Oxford University Press, 1968

[8] Fukunaga K. Introduction to Statistical Recognition. New York: Academic Press, 1990

[9] Graham M, Kevrekidis I. Alternative approaches to the Karhunen-Loève decomposition for model reduction and data analysis. Comput Chem Eng, 1996, 20: 495-506

[10] Holmes P, Lumley J L, Berkooz G. Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge: Cambridge University Press, 1996

[11] Jolliffe I T. Principal Component Analysis. Berlin: Springer-Verlag, 2002

[12] Joslin R D, Gunzburger M D, Nicolaides R, Erlebacher G, Hussaini M Y. A selfcontained automated methodology for optimal flow control validated for transition delay. AIAA Journal, 1997, 35: 816-824

[13] Kunisch K, Volkwein S. Galerkin proper orthogonal decomposition methods for parabolic problems. Numer Math, 2001, 90: 117-148

[14] Kunisch K, Volkwein S. Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J Numer Anal, 2002, 40(2): 492-515

[15] Kunisch K, Volkwein S. Proper orthogonal decomposition for optimality systems. ESAIM: Math Model Numer Anal, 2008, 42(1): 1-23

[16] Lanczos C. Trigonometric interpolation of empirical and analytical functions. J Math Phys, 1938, 17: 123-199

[17] Li H R, Luo Z D, Chen J. Numerical simulation based on proper orthogonal decomposition for two-dimensional solute transport problems. Appl Math Model, 2011, 35(5): 2489-2498

[18] Lumley J L. Coherent structures in turbulence. In: Meyer R E, ed. Transition and Turbulence. Proceedings of the Symposium on Transition and Turbulence in Fluids, Madison, WI, October 13-15, 1980. New York: Academic Press, 1981, 215-242

[19] Luo Z D, Chen J, Navon I M, Yang X Z. Mixed finite element formulation and error estimates based on proper orthogonal decomposition for the non- stationary Navier-Stokes equations. SIAM J Numer Anal, 2008, 47(1): 1-19

[20] Luo Z D, Chen J, Navon I M, Zhu J. An optimizing reduced PLSMFE formulation for non-stationary conduction-convection problems. Int J Numer Meth Fluids, 2009, 60(4): 409-436

[21] Luo Z D, Chen J, Sun P, Yang X Z. Finite element formulation based on proper orthogonal decomposition for parabolic equations. Sci China Ser A: Math, 2009, 52(3): 585-596

[22] Luo Z D, Chen J, Zhu J, Wang R W, Navon I M. An optimizing reduced order FDS for the tropical Pacific Ocean reduced gravity model. Int J Numer Meth Fluids, 2007, 55(2): 143-161

[23] Luo Z D, Du J, Xie Z H, Guo Y. A reduced stabilized mixed finite element formulation based on proper orthogonal decomposition for the no-stationary Navier-Stokes equations. Int J Numer Meth Eng, 2011, 88(1): 31-46

[24] Luo Z D, Li H, Zhou Y J, Huang X M. A reduced FVE formulation based on POD method and error analysis for two-dimensional viscoelastic problem. J Math Anal Appl, 2012, 385: 310-321

[25] Luo Z D, Li H, Zhou Y J, Xie Z H. A reduced finite element formulation and error estimates based on POD method for two-dimensional solute transport problems. J Math Anal Appl, 2012, 385: 371-383

[26] Luo Z D, Ou Q L, Xie Z X. A reduced finite difference scheme and error estimates based on POD method for the non-stationary Stokes equation. Appl Math Mech, 2011, 32(7): 847-858

[27] Luo Z D, Wang R W, Zhu J. Finite difference scheme based on proper orthogonal decomposition for the non-stationary Navier-Stokes equations. Sci China Ser A: Math, 2007, 50(8): 1186-1196

[28] Luo Z D, Xie Z H, Chen J. A reduced MFE formulation based on POD for the nonstationary conduction-convection problems. Acta Math Sci Ser B Engl Ed, 2011, 31(5): 1765-1785

[29] Luo Z D, Xie Z H, Shang Y Q, Chen J. A reduced finite volume element formulation and numerical simulations based on POD for parabolic equations. J Comput Appl Math, 2011, 235(8): 2098-2111

[30] Luo Z D, Yang X Z, Zhou Y J. A reduced finite difference scheme based on singular value decomposition and proper orthogonal decomposition for Burgers equation. J Comput Appl Math, 2009, 229(1): 97-107

[31] Luo Z D, Zhou Y J, Yang X Z. A reduced finite element formulation based on proper orthogonal decomposition for Burgers equation. Appl Numer Math, 2009, 59(8): 1933-1946

[32] Luo Z D, Zhu J, Wang R W, Navon I M. Proper orthogonal decomposition approach and error estimation of mixed finite element methods for the tropical Pacific Ocean reduced gravity model. Comput Meth Appl Mech Eng, 2007, 196(41-44): 4184-4195

[33] Rajaee M, Karlsson S K F, Sirovich L. Low dimensional description of free sheer flow coherent structures and their dynamical behavior. J Fluid Mech, 1994, 258: 1401-1402

[34] Selten F. Baroclinic empirical orthogonal functions as basis functions in an atmospheric model. J Atmospheric Sci, 1997, 54: 2100-2114

[35] Shvartsman S, Kevrekisis I. Low-dimensional approximation and control of periodic solutions in spatially extended systems. Phys Rev E, 1998, 58(3): 361-368

[36] Sirovich L. Turbulence and the dynamics of coherent sructures: part I–III. Quart Appl Math, 1987, 45(3): 561-590

[37] Sun P, Luo Z D, Zhou Y J. Some reduced finite difference schemes based on a proper orthogonal decomposition technique for parabolic equations. Appl Numer Math, 2010, 60(1-2): 154-164

[38] Trefethen L N. Spectral Method in MATLAB. Philadephia: SIAM, 2000

[39] Trültzsch F, Volkwein S. POD a-posteriori error estimates for linear-quadratic optimal control problems. Comput Optim Appl, 2009, 44(1): 83-115

[40] Weideman J A C, Reddy S C. A Matlab differentiation matrix suite. ACM Trans Math Software, 2000, 26: 465-511

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