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Reduced-order extrapolation spectral-finite difference scheme based on POD method and error estimation for three-dimensional parabolic equation. (English) Zbl 1327.65210

Summary: 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 reduced-order 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.

MSC:

65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
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[1] Afanasiev K, Hinze M. Adaptive control of a wake flow using proper orthogonal decomposition. Lect Notes Pure Appl Math, 2001, 216: 317-332 · Zbl 1013.76028
[2] Algazi V, Sakrison D. On the optimality of Karhunen-Loève expansion. IEEE Trans Inform Theory, 1969, 15: 319-321 · Zbl 0175.46602 · doi:10.1109/TIT.1969.1054286
[3] Arian, E.; Fahl, M.; Sachs, E. W., Trust-region proper orthogonal decomposition models by optimization method, 3300-3305 (2002) · doi:10.1109/CDC.2002.1184383
[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 · Zbl 0643.76066
[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 · Zbl 1161.86002 · doi:10.1016/j.camwa.2006.11.012
[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 · Zbl 1370.86002 · doi:10.1002/fld.1365
[7] Fox L, Parker I B. Chebyshev Polynomials in Numerical Analysis. Oxford: Oxford University Press, 1968 · Zbl 0153.17502
[8] Fukunaga K. Introduction to Statistical Recognition. New York: Academic Press, 1990 · Zbl 0711.62052
[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 · doi:10.1016/0098-1354(95)00040-2
[10] Holmes P, Lumley J L, Berkooz G. Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge: Cambridge University Press, 1996 · Zbl 0890.76001 · doi:10.1017/CBO9780511622700
[11] Jolliffe I T. Principal Component Analysis. Berlin: Springer-Verlag, 2002 · Zbl 1011.62064
[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 · Zbl 0901.76067 · doi:10.2514/2.7452
[13] Kunisch K, Volkwein S. Galerkin proper orthogonal decomposition methods for parabolic problems. Numer Math, 2001, 90: 117-148 · Zbl 1005.65112 · doi:10.1007/s002110100282
[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 · Zbl 1075.65118 · doi:10.1137/S0036142900382612
[15] Kunisch K, Volkwein S. Proper orthogonal decomposition for optimality systems. ESAIM: Math Model Numer Anal, 2008, 42(1): 1-23 · Zbl 1141.65050 · doi:10.1051/m2an:2007054
[16] Lanczos C. Trigonometric interpolation of empirical and analytical functions. J Math Phys, 1938, 17: 123-199 · Zbl 0020.01301
[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 · Zbl 1217.74082 · doi:10.1016/j.apm.2010.11.064
[18] Lumley, J. L.; Meyer, R. E (ed.), Coherent structures in turbulence, 215-242 (1981), New York · Zbl 0486.76073
[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 · Zbl 1199.35225 · doi:10.1137/070689498
[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 · Zbl 1161.76032 · doi:10.1002/fld.1900
[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 · Zbl 1183.65122 · doi:10.1007/s11425-008-0125-9
[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 · Zbl 1205.86007 · doi:10.1002/fld.1452
[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 · Zbl 1242.76129 · doi:10.1002/nme.3161
[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 · Zbl 1368.74071 · doi:10.1016/j.jmaa.2011.06.057
[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 · Zbl 1243.65123 · doi:10.1016/j.jmaa.2011.06.051
[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 · Zbl 1237.65126 · doi:10.1007/s10483-011-1464-9
[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 · Zbl 1134.35387 · doi:10.1007/s11425-007-0081-9
[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 · Zbl 1249.65199
[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 · Zbl 1227.65076 · doi:10.1016/j.cam.2010.10.008
[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 · Zbl 1179.65111 · doi:10.1016/j.cam.2008.10.026
[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 · Zbl 1169.65096 · doi:10.1016/j.apnum.2008.12.034
[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 · Zbl 1173.76348 · doi:10.1016/j.cma.2007.04.003
[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 · Zbl 0800.76190 · doi:10.1017/S0022112094003228
[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 · doi:10.1103/PhysRevE.58.361
[36] Sirovich L. Turbulence and the dynamics of coherent sructures: part I-III. Quart Appl Math, 1987, 45(3): 561-590 · Zbl 0676.76047
[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 · Zbl 1193.65159 · doi:10.1016/j.apnum.2009.10.008
[38] Trefethen L N. Spectral Method in MATLAB. Philadephia: SIAM, 2000 · Zbl 0953.68643 · doi:10.1137/1.9780898719598
[39] Trültzsch F, Volkwein S. POD a-posteriori error estimates for linear-quadratic optimal control problems. Comput Optim Appl, 2009, 44(1): 83-115 · Zbl 1189.49050 · doi:10.1007/s10589-008-9224-3
[40] Weideman J A C, Reddy S C. A Matlab differentiation matrix suite. ACM Trans Math Software, 2000, 26: 465-511 · doi:10.1145/365723.365727
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