Optimized finite difference iterative scheme based on POD technique for 2D viscoelastic wave equation

doi:10.1007/s10483-017-2288-8

Applied Mathematics and Mechanics (English Edition) ›› 2017, Vol. 38 ›› Issue (12): 1721-1732.doi: https://doi.org/10.1007/s10483-017-2288-8

Optimized finite difference iterative scheme based on POD technique for 2D viscoelastic wave equation

Hong XIA¹, Zhendong LUO²

1. School of Control and Computer Engineering, North China Electric Power University, Beijing 102206, China;
2. School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China

收稿日期:2017-05-01 修回日期:2017-07-26 出版日期:2017-12-01 发布日期:2017-12-01
通讯作者: Zhendong LUO E-mail:zhdluo@ncepu.edu.cn
基金资助:
Project supported by the National Natural Science Foundation of China (No. 11671106) and the Fundamental Research Funds for the Central Universities (No. 2016MS33)

Optimized finite difference iterative scheme based on POD technique for 2D viscoelastic wave equation

Hong XIA¹, Zhendong LUO²

1. School of Control and Computer Engineering, North China Electric Power University, Beijing 102206, China;
2. School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China

Received:2017-05-01 Revised:2017-07-26 Online:2017-12-01 Published:2017-12-01
Contact: Zhendong LUO E-mail:zhdluo@ncepu.edu.cn
Supported by:
Project supported by the National Natural Science Foundation of China (No. 11671106) and the Fundamental Research Funds for the Central Universities (No. 2016MS33)

摘要/Abstract

摘要： This study develops an optimized finite difference iterative (OFDI) scheme for the two-dimensional (2D) viscoelastic wave equation. The OFDI scheme is obtained using a proper orthogonal decomposition (POD) method. It has sufficiently high accuracy with very few unknowns for the 2D viscoelastic wave equation. Existence, stability, and convergence of the OFDI solutions are analyzed. Numerical simulations verify efficiency and feasibility of the proposed scheme.

关键词: censored data, a.s. convergence, asymptotic normality, K-M estimator, convergence, optimized finite difference iterative (OFDI) scheme, proper orthogonal decomposition (POD), viscoelastic wave equation, existence, numerical simulation, stability

Abstract: This study develops an optimized finite difference iterative (OFDI) scheme for the two-dimensional (2D) viscoelastic wave equation. The OFDI scheme is obtained using a proper orthogonal decomposition (POD) method. It has sufficiently high accuracy with very few unknowns for the 2D viscoelastic wave equation. Existence, stability, and convergence of the OFDI solutions are analyzed. Numerical simulations verify efficiency and feasibility of the proposed scheme.

Key words: censored data, a.s. convergence, asymptotic normality, K-M estimator, stability, numerical simulation, convergence, viscoelastic wave equation, proper orthogonal decomposition (POD), existence, optimized finite difference iterative (OFDI) scheme

中图分类号:

Hong XIA, Zhendong LUO. Optimized finite difference iterative scheme based on POD technique for 2D viscoelastic wave equation[J]. Applied Mathematics and Mechanics (English Edition), 2017, 38(12): 1721-1732.

参考文献

[1] Gurtin, M. and Pipkin, A. A general theory of heat conduction with finite wave speeds. Archive for Rational Mechanics and Analysis, 31, 113-126(1968)
[2] Lin, Y. P. A mixed boundary problem describing the propagation of disturbances in viscous media solution for quasi-linear equations. Journal of Mathematical Analysis and Applications, 135, 644-653(1988)
[3] Suveika, I. V. Mixed problems for an equation describing the propagation of disturbances in viscous media. Journal of Differential Equations, 19, 337-347(1982)
[4] Raynal, M. L. On Some Nonlinear Problems of Diffusion in Volterra Equations, Springer, Berlin (1979)
[5] Yuan, Y. R. Finite difference method and analysis for three-dimensional semiconductor device of heat conduction. Science China Mathematics, 39, 21-32(1996)
[6] Yuan, Y. R. and Wang, H. Error estimates for the finite element methods of nonlinear hyperbolic equations. Journal of System Science and Mathematical Science, 5, 161-171(1985)
[7] Cannon, J. R. and Lin, Y. A priori L² error estimates for finite-element methods for nonlinear diffusion equations with memory. SIAM Journal on Numerical Analysis, 27, 595-607(1999)
[8] Li, H., Zhao, Z. H., and Luo, Z. D. A space-time continuous finite element method for 2D viscoelastic wave equation. Boundary Value Problems, 2016, 1-17(2016)
[9] Sirovich, L. Turbulence, the dynamics of coherent structures. Quarterly of Applied Mathematics, 45, 561-590(1987)
[10] Holmes, P., Lumley, J. L., and Berkooz, G. Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, Cambridge (1996)
[11] 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)
[12] Ly, H. V. and Tran, H. T. Proper orthogonal decomposition for flow calculations and optimal control in a horizontal CVD reactor. Quarterly of Applied Mathematics, 60, 631-656(2002)
[13] 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)
[14] 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)
[15] Luo, Z. D., Chen, J., Navon, I. M., and Yang, X. Z. 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)
[16] Luo, Z. D., Yang, X. Z., 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)
[17] 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)
[18] 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 problems. Journal of Computational and Applied Mathematics, 235, 2098-2111(2011)
[19] 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 non-stationary Navier-Stokes equations. International Journal for Numerical Methods in Engineering, 88, 31-46(2011)
[20] 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)
[21] Ghosh, R. and Joshi, Y. Error estimation in POD-based dynamic reduced-order thermal modeling of data centers. International Journal of Heat and Mass Transfer, 57, 698-707(2013)
[22] Stefanescu, R., Sandu, A., and Navon, I. M. Comparison of POD reduced order strategies for the nonlinear 2D shallow water equations. International Journal for Numerical Methods in Fluids, 76, 497-521(2014)
[23] Luo, Z. D. and Teng, F. Reduced-order proper orthogonal decomposition extrapolating finite volume element format for two-dimensional hyperbolic equations. Applied Mathematics and Mechanics, 38, 289-310(2017)
[24] Dimitriu, G., Stefanescu, R., and Navon, I. M. POD-DEIM approach on dimension reduction of a multi-species host-parasitoid system. Annals of the Academy of Romanian Scientists, 7, 173-188(2015)
[25] Luo, Z. D. A POD-based reduced-order finite difference extrapolating model for the non-stationary incompressible Boussinesq equations. Advances in Difference Equations, 2014, 1-18(2014)
[26] An, J., Luo, Z. D., Li, H., and Sun, P. Reduced-order extrapolation spectral-finite difference scheme based on POD method and error estimation for three-dimensional parabolic equation. Frontiers of Mathematics in China, 10, 1025-1040(2015)
[27] Luo, Z. D. and Gao, J. Q. A POD-based reduced-order finite difference time-domain extrapolating scheme for the 2D Maxwell equations in a lossy medium. Journal of Mathematical Analysis and Applications, 444, 433-451(2016)
[28] Luo, Z. D., Jin, S. J., and Chen, J. A reduced-order extrapolation central difference scheme based on POD for two dimensional fourth-order hyperbolic equations. Applied Mathematics and Computation, 289, 396-408(2016)
[29] Zhang, W. S. Finite Difference Methods for Partial Differential Equations in Science Computation, Higher Education Press, Beijing (2006)
[30] Chung, T. Computational Fluid Dynamics, Cambridge University Press, Cambridge (2002)
[31] He, Y. N. and Sun, W. W. Stability and convergence of the Crank-Nicolson/Adams-Bashforth scheme for the time-dependent Navier-Stokes equations. SIAM Journal on Numerical Analysis, 45, 837-869(2007)
[32] He, Y. N. The Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data. Mathematics of Computation, 77, 2097-2124(2008)

Optimized finite difference iterative scheme based on POD technique for 2D viscoelastic wave equation

Optimized finite difference iterative scheme based on POD technique for 2D viscoelastic wave equation

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

编辑推荐

Metrics

本文评价

[1]	N. HUMNEKAR, D. SRINIVASACHARYA. Influence of variable viscosity and double diffusion on the convective stability of a nanofluid flow in an inclined porous channel[J]. Applied Mathematics and Mechanics (English Edition), 2024, 45(3): 563-580.
[2]	Haitao LI, Tianyu ZHENG, Weiyang QIN, Ruilan TIAN, Hu DING, J. C. JI, Liqun CHEN. Theoretical and experimental study of a bi-stable piezoelectric energy harvester under hybrid galloping and band-limited random excitations[J]. Applied Mathematics and Mechanics (English Edition), 2024, 45(3): 461-478.
[3]	Haiyang WU, Jiangfeng LOU, Biao ZHANG, Yuntong DAI, Kai LI. Stability analysis of a liquid crystal elastomer self-oscillator under a linear temperature field[J]. Applied Mathematics and Mechanics (English Edition), 2024, 45(2): 337-354.
[4]	Pan WANG, Xiangcheng HAN, Weibin WEN, Baolin WANG, Jun LIANG. Galerkin-based quasi-smooth manifold element (QSME) method for anisotropic heat conduction problems in composites with complex geometry[J]. Applied Mathematics and Mechanics (English Edition), 2024, 45(1): 137-154.
[5]	Chen ZHAO, Zhenli CHEN, C. T. MUTASA, Dong LI. Effects of layer interactions on instantaneous stability of finite Stokes flows[J]. Applied Mathematics and Mechanics (English Edition), 2024, 45(1): 69-84.
[6]	Haoyang LI, Weijian LIU, Yuhong DONG. A rescaling algorithm for multi-relaxation-time lattice Boltzmann method towards turbulent flows with complex configurations[J]. Applied Mathematics and Mechanics (English Edition), 2023, 44(9): 1597-1612.
[7]	A. V. ROŞCA, N. C. ROŞCA, I. POP. Three-dimensional mixed convection stagnation-point flow past a vertical surface with second-order slip velocity[J]. Applied Mathematics and Mechanics (English Edition), 2023, 44(4): 641-652.
[8]	M. HAMID, M. USMAN, Zhenfu TIAN. Computational analysis for fractional characterization of coupled convection-diffusion equations arising in MHD flows[J]. Applied Mathematics and Mechanics (English Edition), 2023, 44(4): 669-692.
[9]	Peng WANG, Shaopu YANG, Yongqiang LIU, Pengfei LIU, Xing ZHANG, Yiwei ZHAO. Research on hunting stability and bifurcation characteristics of nonlinear stochastic wheelset system[J]. Applied Mathematics and Mechanics (English Edition), 2023, 44(3): 431-446.
[10]	Xiang KANG, Yujin TONG, Wei WU, Xingzhe WANG. Transient multi-physics behavior of an insert high temperature superconducting no-insulation coil in hybrid superconducting magnets with inductive coupling[J]. Applied Mathematics and Mechanics (English Edition), 2023, 44(2): 255-272.
[11]	Pei ZHANG, P. SCHIAVONE, Hai QING. Dynamic stability analysis of porous functionally graded beams under hygro-thermal loading using nonlocal strain gradient integral model[J]. Applied Mathematics and Mechanics (English Edition), 2023, 44(12): 2071-2092.
[12]	J. R. FERNÁNDEZ, R. QUINTANILLA. A higher-order porous thermoelastic problem with microtemperatures[J]. Applied Mathematics and Mechanics (English Edition), 2023, 44(11): 1911-1926.
[13]	Zilong GUO, Qiao NI, Wei CHEN, Huliang DAI, Lin WANG. Dynamic analysis and regulation of the flexible pipe conveying fluid with a hard-magnetic soft segment[J]. Applied Mathematics and Mechanics (English Edition), 2022, 43(9): 1415-1430.
[14]	Si YUAN, Quan YUAN. Condensed Galerkin element of degree m for first-order initial-value problem with O(h^2m+2) super-convergent nodal solutions[J]. Applied Mathematics and Mechanics (English Edition), 2022, 43(4): 603-614.
[15]	P. Z. S. PAZ, F. R. CUNHA, Y. D. SOBRAL. Stability of plane-parallel flow of magnetic fluids under external magnetic flelds[J]. Applied Mathematics and Mechanics (English Edition), 2022, 43(2): 295-310.