DOI QR코드

DOI QR Code

NUMERICAL SOLUTIONS OF BURGERS EQUATION BY REDUCED-ORDER MODELING BASED ON PSEUDO-SPECTRAL COLLOCATION METHOD

  • Received : 2015.03.13
  • Accepted : 2015.06.03
  • Published : 2015.06.25

Abstract

In this paper, a reduced-order modeling(ROM) of Burgers equations is studied based on pseudo-spectral collocation method. A ROM basis is obtained by the proper orthogonal decomposition(POD). Crank-Nicolson scheme is applied in time discretization and the pseudo-spectral element collocation method is adopted to solve linearlized equation based on the Newton method in spatial discretization. We deliver POD-based algorithm and present some numerical experiments to show the efficiency of our proposed method.

Acknowledgement

Supported by : National Research Foundation of Korea(NRF)

References

  1. K. Kunisch and S. Volkwein, Control of the Burgers Equation by a Reduced-Order Approach Using Proper Orthogonal Decomposition, Jour. Optim. Theory and Appl., 102(2) (1999) 345-371. https://doi.org/10.1023/A:1021732508059
  2. K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic equations, Numer. Math. 90 (2001), 117-148 . https://doi.org/10.1007/s002110100282
  3. J. Burkardt, M. Gunzburger and H.-C. Lee, Centroidal Voronoi tessellation-based reduced-order modeling of complex systems, SIAM J. SCI. COMPUT. Vol. 28 (2006), No. 2, pp. 459-484. https://doi.org/10.1137/5106482750342221x
  4. J. Burkardt, M. Gunzburger and H.-C. Lee, POD and CVT-based reduced-order modeling of Navier-Stokes flows, Comput. Methods Appl. Mech. Engrg. 196 (2006), 337-355. https://doi.org/10.1016/j.cma.2006.04.004
  5. O. E. Mehmet, Fuzzy Boundary Control of 2D Burgers Equation with an Observer, Proceedings of the 2005 IEEE Conference on Control Applications, Toronto, Canada, August 28-31, (2005).
  6. E. Aksan, A numerical solution of Burgers equation by finite element method constructed on the method of discretization in time, Appl. Math. Comp. 170 (2005), 895-904. https://doi.org/10.1016/j.amc.2004.12.027
  7. E. Aksan, A. Ozdes and T. Ozis, A numerical solution of Burgers equation based on least squares approximation, Appl. Math. Comp. 176 (2006), 270-279. https://doi.org/10.1016/j.amc.2005.09.045
  8. S. Kutluay, A. Esen and I. Dag, Numerical solutions of the Burgers equation by the least-squares quadratic B-spline finite element method, J. Comput. Appl. Math. 167 (2004), 21-33. https://doi.org/10.1016/j.cam.2003.09.043
  9. M. K. Kadalbajoo and A. Awasthi, A numerical method based on Crank-Nicolson scheme for Burgers' equation, Appl. Math. Comp. 182 (2006), 1430-1442. https://doi.org/10.1016/j.amc.2006.05.030
  10. H.-C. Lee, S. Lee and G. Piao Reduced-order modeling of Burgers equations based on Centroidal Voronoi Tessellation, Inter. J. Numer. Anal. Model., 4(3-4) (2007), 559-583.
  11. M. Gunzburger, J. Peterson and J. Shadid, Reduced-order modeling of time-dependent PDEs with multiple parameters in the boundary data, Comput. Meth. Appl. Mech. Engrg. 196 (2007), 1030-1047. https://doi.org/10.1016/j.cma.2006.08.004
  12. Y. C. Liang, H. P. Lee, S. P. Lim, W. Z. Lin, K. H. Lee and C. G. Wu, Proper orthogonal decomposition and its application-part1: Theory, J. Sound and Vibration, 252(3) (2002), 527-544. https://doi.org/10.1006/jsvi.2001.4041
  13. B.-C. Shin and J.-H. Jung, Spectral collocation and radial basis function methods for one-dimensional interface problems, Appl. Numer. Math. 61 (2011), 911-928. https://doi.org/10.1016/j.apnum.2011.03.005
  14. D. Funaro, Spectral Elements for Transport-Dominated Equations, Springer, (1997).
  15. S. D. Kim, H.-C. Lee and B.-C. Shin, Pseudospectral least-squares method for the second-order elliptic boundary value problem, SIAM J. Numer Anal. 41(4) (2003), 1370-1387. https://doi.org/10.1137/S0036142901398234
  16. A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equation, Springer-Verlag, Berlin Heidelberg, (1994).
  17. Q. Du, V. Faber and M. Gunzburger, Centroidal Voronoi Tessellations: Applications and Algorithms, SIAM Rev. 41(4) (1999), 637-676. https://doi.org/10.1137/S0036144599352836