A Reduced-Order Extrapolating Finite Difference Iterative Scheme for 2D Generalized Nonlinear Sine-Gordon Equation
Applied and Computational Mathematics
Volume 7, Issue 1, February 2018, Pages: 19-25
Received: Dec. 17, 2017;
Accepted: Jan. 2, 2018;
Published: Jan. 18, 2018
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Hong Xia, School of Control and Computer Engineering, North China Electric Power University, Beijing, China
Fei Teng, School of Control and Computer Engineering, North China Electric Power University, Beijing, China
Zhendong Luo, School of Mathematics and Physics, North China Electric Power University, Beijing, China
In this study, a reduced-order extrapolating finite difference iterative (ROEFDI) scheme holding sufficiently high accuracy but containing very few degrees of freedom for the two-dimensional (2D) generalized nonlinear Sine-Gordon equation is built via the proper orthogonal decomposition. The stability and convergence of the ROEFDI solutions are analyzed. And the feasibility and effectiveness of the ROEFDI scheme are verified by numerical simulations. This means that the ROEFDI scheme is effective and feasible for finding the numerical solutions of the 2D generalized nonlinear Sine-Gordon equation.
A Reduced-Order Extrapolating Finite Difference Iterative Scheme for 2D Generalized Nonlinear Sine-Gordon Equation, Applied and Computational Mathematics.
Vol. 7, No. 1,
2018, pp. 19-25.
E. Bour, “Théorie de la déformation des surfaces,” J. Ecole Imperiale Polytechnique, 1862, 19, 1–48.
J. Frenkel and T. Kontorova, “On the theory of plastic deformation and twinning,” Izvestiya Akademii Nauk SSSR, Seriya Fizicheskaya, 1939, 1 (3), 1607–1614.
A. E. Koshelev, “Stability of dynamic coherent states in intrinsic Josephson-junction stacks near internal cavity resonance,” Phys. Rev., 2010, 82, 174512–174526.
B. Batiha, M. S. M. Noorani and I. Hashim, “Approximate analytical solution of the coupled sine-Gordon equation using the variational iteration method,” Physica Scripta, 2007, 76 (5), 445–448.
H. Jafari, R. Soltani and C. M. Khalique, “Exact solutions of two nonlinear partial differential equations by using the first integral method,” Boundary Value Problems, 2013, 2013 (1), 1–9.
S. F. Zhou, “Dimension of the global attractor for the damped Sine-Gordon equation,” Acta Mathematica sinica, 1996, 39 (5), 597–601.
T. Chung, “Computational Fluid Dynamics”, Cambridge University Press, Cambridge, 2002.
L. Sirovich, “Turbulence, the dynamics of coherent structures: part I-III,” Quart. Appl. Math., 1987, 45, 561–590.
J. S. Hesthaven, G. Rozza and B. Stamm, “Certified Reduced Basis Methods for Parametrized Partial Differential Equations,” Springer International Publishing, 2016.
H. Xia and Z. D. Luo, “A stabilized MFE reduced-order extrapolation model based on POD for the 2D unsteady conduction-convection problem,” Journal of Inequalities and Applications, 2017, 2017 (124), 1–17.
P. Benner， A. Cohen, M. Ohlberger and A. K. Willcox, “Model Reduction and Approximation: Theory and Algorithm,” Computational Science & Engineering, SIAM, 2017.
A. Quarteroni, A. Manzoni and F. Negri, “Reduced Basis Methods for Partial Differential Equations,” Springer International Publishing, 2016.
Z. D. Luo and S. J. Jin, “A reduced-order extrapolation spectral-collocation scheme based on POD method for 2D second-order hyperbolic equations,” Mathematical Modelling and Analysis，2017, 22 (5), 569–586.
Z. D. Luo, F. Teng and J. Chen, “A POD-based reduced-order Crank-Nicolson finite volume element extrapolating algorithm for 2D Sobolev equations,” Mathematics and Computers in Simulation, 2018，146, 118–133.
Z. D. Luo and F. Teng, “An optimized SPDMFE extrapolation approach based on the POD technique for 2D viscoelastic wave equation,” Boundary Value Problems, 2017, 2017 (6), 1–20.
Z. D. Luo and F. Teng, “Reduced-order proper orthogonal decomposition extrapolating finite volume element format for two-dimensional hyperbolic equations,” Applied Mathematics and Mechanics, 2017, 38 (2), 289–310.
Z. D. Luo, “A POD-based reduced-order TSCFE extrapolation iterative format for two-dimensional heat equations,” Boundary Value Problems, 2015, 2015 (59), 1–15.
Z. D. Luo, “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, 2015, 20 (4), 1189–1212.
Z. D. Luo, “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, 2015, 425 (1), 259–280.
R. Stefanescu, A. Sandu and I. M. Navon, “Comparison of POD reduced order strategies for the nonlinear 2D shallow water equations,” International Journal for Numerical Methods in Fluids, 2014, 76 (8), 497–521.
G. Dimitriu, R. Stefanescu and I. M. Navon, “POD-DEIM approach on dimension reduction of a multi-specices host-parasition system,” Ann. Acad. Rom. Sci. Ser. Math. Appl., 2015, 7 (1), 173–188.
Z. D. Luo, “A POD-based reduced-order finite difference extrapolating model for the non-stationary incompressible Boussinesq equations,” Adv. Difference Eq., 2014, 2014 (272), 1–18.
Z. D. Luo, S. J. Jin and J. Chen, “A reduced-order extrapolation central difference scheme based on POD for two dimensional fourth-order hyperbolic equations,” Appl. Math. Comput., 2016, 289, 396–408.
Z. D. Luo, “A POD-based reduced-order TSCFE extrapolation iterative format for two-dimensional heat equations,” Bound. Value Probl., 2015, 59 (1), 1–15.
J. An, Z. D. Luo, H. Li and P. Sun, “Reduced-order extrapolation spectral-finite difference scheme based on POD method and error estimation for three-dimensional parabolic equation,” Front. Math. China, 2015, 10 (5), 1025–1040.
W. S. Zhang, “Finite Difference Methods for Partial Differential Equations in Science Computation,” Higher Education Press, Beijing, 2006.