Reduced-order proper orthogonal decomposition extrapolating finite volume element format for two-dimensional hyperbolic equations

doi:10.1007/s10483-017-2162-9

Abstract

Abstract:

This paper is concerned with establishing a reduced-order extrapolating finite volume element(FVE) format based on proper orthogonal decomposition(POD) for two-dimensional(2D) hyperbolic equations. For this purpose, a semi discrete variational format relative time and a fully discrete FVE format for the 2D hyperbolic equations are built, and a set of snapshots from the very few FVE solutions are extracted on the first very short time interval. Then, the POD basis from the snapshots is formulated, and the reduced-order POD extrapolating FVE format containing very few degrees of freedom but holding sufficiently high accuracy is built. Next, the error estimates of the reduced-order solutions and the algorithm procedure for solving the reduced-order format are furnished. Finally, a numerical example is shown to confirm the correctness of theoretical conclusions. This means that the format is efficient and feasible to solve the 2D hyperbolic equations.

Key words: error estimate, numerical simulation, reduced-order finite volume element(FVE) extrapolating format, hyperbolic equation, proper orthogonal decomposition(POD)

2010 MSC Number:

O241.82

Zhendong LUO, Fei TENG. Reduced-order proper orthogonal decomposition extrapolating finite volume element format for two-dimensional hyperbolic equations. Applied Mathematics and Mechanics (English Edition), 2017, 38(2): 289-310.

TrendMD

References

[1] Holmes, P., Lumley, J. L., and Berkooz, G. Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, Cambridge (1996)
[2] Fukunaga, K. Introduction to Statistical Recognition, Academic Press, New York (1990)
[3] Jolliffe, I. T. Principal Component Analysis, Springer-Verlag, Berlin (2002)
[4] Aubry, N., Holmes, P., Lumley, J. L., and Stone, E. The dynamics of coherent structures in the wall region of a turbulent boundary layer. Journal of Fluid Mechanics, 192, 115-173(1988)
[5] Berkooz, G., Holmes, P., and Lumley, J. L. The proper orthogonal decomposition in analysis of turbulent flows. Annual Review of Fluid Mechanics, 25, 539-575(1993)
[6] Cazemier, W., Verstappen, R. W. C. P., and Veldman, A. E. P. Proper orthogonal decomposition and low-dimensional models for driven cavity flows. Physics of Fluids, 10, 1685-1699(1998)
[7] Jones, W. P. and Menziest, K. R. Analysis of the cell-centred finite volume method for the diffusion equation. Journal of Computational Physics, 165, 45-68(2000)
[8] Ko, J., Kurdila, A. J., Redionitis, O. K., and Yue, X. Synthetic jets, their reduced-order modeling and applications to flow control. 37 Aerospace Sciences Meeting and Exhibit, American Institute of Aeronautics and Astronautics, Reno (1999)
[9] Lumley, J. L. Coherent Structures in Turbulence (ed. Meyer, R. E.), Transition and Turbulence, Academic Press, New York, 215-242(1981)
[10] Ly, H. V. and Tran, H. T. Proper orthogonal decomposition for flow calculations and optimal control in a horizontal CVD reactor. Quartterly of Applied Mathematics, 60, 631-656(2002)
[11] Moin, P. and Moser, R. D. Characteristic-eddy decomposition of turbulence in channel. Journal of Fluid Mechanics, 200, 417-509(1989)
[12] Rajaee, M., Karlsson, S. K. F., and Sirovich, L. Low dimensional description of free shear flow coherent structures and their dynamical behavior. Journal of Fluid Mechanics, 258, 1401-1402(1994)
[13] Roslin, R. D., Gunzburger, M. D., Nicolaides, R. A., Erlebacher, G., and Hussaini, M. Y. A selfcontained automated methodology for optimal flow control validated for transition delay. AIAA Journal, 35, 816-824(1997)
[14] Selten, F. Baroclinic empirical orthogonal functions as basis functions in an atmospheric model. Journal of the Atmospheric Sciences, 54, 2100-2114(1997)
[15] Sirovich, L. Turbulence and the dynamics of coherent structures, part I-Ⅲ. Quartterly of Applied Mathematics, 45, 561-590(1987)
[16] Kunisch, K. and Volkwein, S. Galerkin proper orthogonal decomposition methods for parabolic problems. Numerische Mathematik, 90, 117-148(2001)
[17] Kunisch, K. and Volkwein, S. Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM Journal on Numerical Analysis, 40, 492-515(2002)
[18] Kunisch, K. and Volkwein, S. Control of Burgers' equation by a reduced-order approach using proper orthogonal decomposition. Journal Optimization Theory and Applications, 102, 345-371(1999)
[19] Ahlman, D., Södelund, F., Jackson, J., Kurdila, A., and Shyy, W. Proper orthogonal decomposition for time-dependent lid-driven cavity flows. Numerical Heat Transfer, Part B:Fundamentals, 42, 285-306(2002)
[20] Luo, Z. D., Ou, Q. L., and Xie, Z. H. A reduced finite difference scheme and error estimates based on POD method for the non-stationary Stokes equation. Applied Mathematics and Mechanics (English Edition), 32(7), 847-858(2011) DOI 10.1007/s10483-011-1464-9
[21] Cao, Y. H., Zhu, J., Luo, Z. D., and Navon, I. M. Reduced order modeling of the upper tropical pacific ocean model using proper orthogonal decomposition. Computers and Mathematics with Applications, 52, 1373-1386(2006)
[22] Luo, Z. D., Du, J., Xie, Z. H., and Guo, Y. A reduced stabilized mixed finite element formulation based on proper orthogonal decomposition for the no-stationary Navier-Stokes equations. International Journal of Numerical Methods in Engineering, 88, 31-46(2011)
[23] Luo, Z. D., Chen, J., Xie, Z. H., An, J., and Sun, P. A reduced second-order time accurate finit element formulation based on POD for parabolic equations. Scientia Sinica Mathematica, 41, 447-460(2011)
[24] Luo, Z. D., Chen, J., Sun, P., and Yang, X. Finite element formulation based on proper orthogonal decomposition for parabolic equations. Science in China Series A:Mathematics, 52, 587-596(2009)
[25] Luo, Z. D., Chen, J., Zhu, J., Wang, R. W., and Navon, I. M. An optimizing reduced-order FDS for the tropical Pacific Ocean reduced gravity model. International Journal of Numerical Methods in Fluids, 55, 143-161(2007)
[26] Luo, Z. D., Wang, R. W., and Zhu, J. Finite difference scheme based on proper orthogonal decomposition for the non-stationary Navier-Stokes equations. Science in China Series A:Mathematics, 50, 1186-1196(2007)
[27] Luo, Z. D., Yang, X., and Zhou, Y. J. A reduced finite difference scheme based on singular value decomposition and proper orthogonal decomposition for Burgers equation. Journal of Computational and Applied Mathematics, 229, 97-107(2009)
[28] Luo, Z. D., Zhou, Y. J., and Yang, X. A reduced finite element formulation based on proper orthogonal decomposition for Burgers equation. Applied Numerical Mathematics, 59, 1933-1946(2009)
[29] Luo, Z. D., Zhu, J., Wang, R. W., and Navon, I. M. Proper orthogonal decomposition approach and error estimation of mixed finite element methods for the tropical Pacific Ocean reduced gravity model. Computer Methods in Applied Mechanics and Engineering, 196, 4184-4195(2007)
[30] Sun, P., Luo, Z. D., and Zhou, Y. J. Some reduced finite difference schemes based on a proper orthogonal decomposition technique for parabolic equations. Applied Numerical Mathematics, 60, 154-164(2010)
[31] Luo, Z. D., Chen, J., Navon, I. M., and Yang, X. Mixed finite element formulation and error estimates based on proper orthogonal decomposition for the non-stationary Navier-Stokes equations. SIAM Journal on Numerical Analysis, 47, 1-19(2008)
[32] Luo, Z. D., Ou, Q. L., Wu, J. R., and Xie, Z. H. A reduced FE formulation based on POD methed for hyperbolic equation. Acta Mathematica Scientia, 32, 1997-2009(2012)
[33] Luo, Z. D., Li, H., Shang, Y. Q., and Fang, Z. A reduced-order LSMFE formulation based on proper orthogonal decomposition for parabolic equations. Finite Elements in Analysis and Design, 60, 1-12(2012)
[34] Luo, Z. D., Li, H., Zhou, Y. J., and Xie, Z. H. A reduced finite element formulation and error estimates based on POD method for two-dimensional solute transport problems. Journal of Mathematical Analysis and Applications, 385, 371-383(2012)
[35] Luo, Z. D., Xie, Z. H., Shang, Y. Q., and Chen, J. A reduced finite volume element formulation and numerical simulations based on POD for parabolic equations. Journal of Computational and Applied Mathematical, 235, 2098-2111(2011)
[36] Luo, Z. D., Li, H., Zhou, Y. J., and Huang, X. A reduced FVE formulation based on POD method and error analysis for two-dimensional viscoelastic problem. Journal of Mathematical Analysis and Applications, 385, 310-321(2012)
[37] Luo, Z. D. A reduced-order SMFVE extrapolation algorithm based on POD technique and CN method for the non-stationary Navier-Stokes equations. Discrete and Continuous Dynamical Systems Series B, 20(4), 1189-1212(2015)
[38] Luo, Z. D. Proper orthogonal decomposition-based reduced-order stabilized mixed finite volume element extrapolating model for the nonstationary incompressible Boussinesq equations. Journal of Mathematical Analysis and Applications, 425, 259-280(2015)
[39] Luo, Z. D., Li, H., Chen, J., and Teng, F. A reduced-order FVE extrapolation algorithm based on proper orthogonal decomposition technique and its error analysis for Sobolev equation. Journal of Industrial and Applied Mathematics, 32(1), 119-142(2015)
[40] Luo, Z. D. A reduced-order extrapolation algorithm based on SFVE method and POD technique for non-stationary Stokes equations. Applied Mathematics and Computation, 247, 976-995(2014)
[41] Cai, Z. and McCormick, S. On the accuracy of the finite volume element method for diffusion equations on composite grid. SIAM Journal on Numerical Analysis, 27, 636-655(1990)
[42] Süli, E. Convergence of finite volume schemes for Poisson's equation on nonuniform meshes. SIAM Journal on Numerical Analysis, 28, 1419-1430(1991)
[43] Bank, R. E. and Rose, D. J. Some error estimates for the box methods. SIAM Journal on Numerical Analysis, 24, 777-787(1987)
[44] Li, R. H., Chen, Z. Y., and Wu, W. Generalized difference methods for differential equations, numerical analysis of finite volume methods, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 226, Marcel Dekker, New York (2000)
[45] Kumar, S., Nataraj, N., and Pani, A. K. Finite volume element method for second order hyperbolic equations. International Journal of Numerical Analysis and Modeling, 5, 132-151(2008)
[46] Adams, R. A. Sobolev Spaces, Academic Press, New York (1975)
[47] Dupont, T. L²-estimates for Galerkin methods for second order hyperbolic equations. SIAM Journal on Numerical Analysis, 10, 880-889(1973)
[48] Pani, A. K. and Sinha, R. K. The efect of spatial quadrature on finite element Galerkin approximation to hyperbolic integro-differential equations. Numerical Functional Analysis Optimization, 19, 1129-1153(1998)
[49] Sinha, R. K. Finite element approximations with quadrature for second-order hyperbolic equations. Numerical Methods for Partial Differential Equations, 18, 537-559(2002)
[50] Luo, Z D. Mixed Finite Element Methods and Application, Chinese Science Press, Beijing (2006)
[51] Brezzi, F. and Fortin, M. Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York (1991)
[52] Ciarlet, P.G. The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, New York (1978)
[53] Li, J. and Chen, Z. X. A new stabilized finite volume method for the stationary Stokes equations. Advances in Computational Mathematics, 30, 141-152(2009)
[54] Shen, L. H., Li, J., and Chen, Z. X. Analysis of a stabilized finite volume method for the transient stationary Stokes equations. International Journal of Numerical Analysis and Modeling, 6, 505-519(2009)
[55] Rudin, W. Functional and Analysis (2nd ed), McGraw-Hill, New York (1973)
[56] Temam, R. Navier-Stokes Equations (3rd ed), North-Holland, Amsterdam/New York (1984)