DOI QR코드

DOI QR Code

PROPER ORTHOGONAL DECOMPOSITION OF DISCONTINUOUS SOLUTIONS WITH THE GEGENBAUER POST-PROCESSING

  • SHIN, BYEONG-CHUN (DEPARTMENT OF MATHEMATICS, CHONNAM NATIONAL UNIVERSITY) ;
  • JUNG, JAE-HUN (DEPARTMENT OF AI & DATA SCIENCE, AJOU UNIVERSITY, DEPARTMENT OF MATHEMATICS, UNIVERSITY AT BUFFALO SUNY)
  • Received : 2019.10.30
  • Accepted : 2019.12.03
  • Published : 2019.12.25

Abstract

The proper orthogonal decomposition (POD) method for time-dependent problems significantly reduces the computational time as it reduces the original problem to the lower dimensional space. Even a higher degree of reduction can be reached if the solution is smooth in space and time. However, if the solution is discontinuous and the discontinuity is parameterized e.g. with time, the POD approximations are not accurate in the reduced space due to the lack of ability to represent the discontinuous solution as a finite linear combination of smooth bases. In this paper, we propose to post-process the sample solutions and re-initialize the POD approximations to deal with discontinuous solutions and provide accurate approximations while the computational time is reduced. For the post-processing, we use the Gegenbauer reconstruction method. Then we regularize the Gegenbauer reconstruction for the construction of POD bases. With the constructed POD bases, we solve the given PDE in the reduced space. For the POD approximation, we re-initialize the POD solution so that the post-processed sample solution is used as the initial condition at each sampling time. As a proof-of-concept, we solve both one-dimensional linear and nonlinear hyperbolic problems. The numerical results show that the proposed method is efficient and accurate.

Acknowledgement

Supported by : National Research Foundation of Korea(NRF)

References

  1. P. Benner, M. Ohlberger, A. T. Patera, G. Rozza, D. C. Sorensen, K. Urban (Eds.), Special Issue: Model Order Reduction of Parameterized Systems, Adv. Comput. Math. 41, 2015.
  2. G. Berkooz, P. Holmes, J. Lumley, The proper orthogonal decomposition in the analysis of turbulent flows, Ann. Rev. Fluid Mech. 25 (1993) 539-575. https://doi.org/10.1146/annurev.fl.25.010193.002543
  3. J. Burkardt, M. Gunzburger, H.-C. Lee, Centroidal Voronoi tessellation-based reduced-order modeling of complex systems, SIAM J. Sci. Comput. 28(2) (2006) 459-484.
  4. J. Burkardt, M. Gunzburger, H.-C. Lee, POD and CVT-based reduced-order modeling of Navier-Stokes flows, Comput. Meth. Appl. Mech. Eng. 196 (2006) 337-355. https://doi.org/10.1016/j.cma.2006.04.004
  5. Q. Du, V. Faber, M. Gunzburger, Centroidal Voronoi tessellations: applications and algorithms, SIAM Rev. 41(4) (1999) 637-676. https://doi.org/10.1137/S0036144599352836
  6. M. Gunzburger, J. Peterson, J. Shadid, Reduced-order modeling of time-dependent PDEs with multiple parameters in the boundary data, Comput. Meth. Appl. Mech. Eng. 196 (2007) 1030-1047. https://doi.org/10.1016/j.cma.2006.08.004
  7. K. Kunisch, S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic equations, Numer. Math. 90 (2001) 117-148. https://doi.org/10.1007/s002110100282
  8. Y. C. Liang, H. P. Lee, S. P. Lim, W. Z. Lin, K. H. Lee, 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
  9. H.-C. Lee, S. Lee, G. Piao, Reduced-order modeling of Burgers equations based on centroidal Voronoi tessellation, Int. J. Numer. Anal. Model 4(3-4) (2007) 559-583.
  10. S. Ravindran, Reduced-order adaptive controllers for fluid flows using POD, SIAM J. Sci. Comput. 15 (2000) 457-478. https://doi.org/10.1023/A:1011184714898
  11. F. Fang, C. C. Pain, I. M. Navon, A. H. Elsheikh, J. Du, D. Xiao, Non-linear Petrov-Galerkin methods for reduced order hyperbolic equations and discontinuous finite element methods, J. Comput. Phy. 234 (2013) 540-559. https://doi.org/10.1016/j.jcp.2012.10.011
  12. D. Gottlieb, C-W.Shu, On the Gibbs phenomenon and its resolution, SIAM Rev. 39 (1997) 644-668. https://doi.org/10.1137/S0036144596301390
  13. D. Gottlieb, C-W. Shu, A.Solomonoff, H. Vandeven, On the Gibbs phenomenon 1: recovering exponential accuracy from the Fourier partial sum of a nonperiodic analytic function, J. Comput. Appl. Math. 43 (1992) 81-92. https://doi.org/10.1016/0377-0427(92)90260-5
  14. D. Gottlieb, C.-W. Shu, On the Gibbs phenomenon V: recovering exponential accuracy from collocation point values of a piecewise analytic function, Numerische Mathematik 71 (1995) 511-526. https://doi.org/10.1007/s002110050155
  15. D. Gottlieb, C.-W. Shu, On the Gibbs phenomenon IV: recovering exponential accuracy in a sub-interval from a Gegenbauer partial sum of a piecewise analytic function, Math. Comput. 64 (1995) 1081-1095.
  16. D. Gottlieb, C.-W. Shu, On the Gibbs phenomenon III: recovering exponential accuracy in a subinterval from a spectral partial sum of a piecewise analytic function, SIAM J. Numer. Anal. 33 (1996) 280-290. https://doi.org/10.1137/0733015
  17. B.-C. Shin, J.-H. Jung, Spectral collocation and radial basis function methods for one-dimensional interface problems, Appl. Numer. Math. 61(8) (2011) 911-928. https://doi.org/10.1016/j.apnum.2011.03.005
  18. C.-W. Shu, S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys. 77 (1988) 439-471. https://doi.org/10.1016/0021-9991(88)90177-5
  19. J. Hesthaven, S. Gottlieb, D. Gottlieb, Spectral Methods for Time-Dependent Problems, Cambridge University Press, Cambridge, 2007.
  20. M. Abramowitz, I. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, tenth ed., New York, 1964.
  21. W.-S. Don, D. Gottlieb, J.-H. Jung, Multi-domain Spectral Method for Supersonic Reactive Flows, J. Comput. Phys. 192(1) (2003) 325-354. https://doi.org/10.1016/j.jcp.2003.07.022
  22. E. Tadmor, J. Tanner, Adaptive filters for piecewise smooth spectral data, IMA J. Num. Anal. 25 (2005) 635-647. https://doi.org/10.1093/imanum/dri026
  23. R. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge UP, Cambridge, 2002.
  24. J.-K. Seo, B.-C. Shin, Numerical solutions of Burgers equation by reduced-order modeling based on pseudospectral collocation method, J. KSIAM 19(2) (2015) 123-135.
  25. R. Archibald, K.W. Chen, A. Gelb, R. Renaut, Improving tissue segmentation of human brain MRI through preprocessing by the Gegenbauer reconstruction method, Neuroimage, 20 (2003) 489-502. https://doi.org/10.1016/S1053-8119(03)00260-X
  26. A. Gelb,, D. Gottlieb, The resolution of the Gibbs phenomenon for "spliced" functions in one and two dimensions, Comput. Math. Appl. 33 (1997) 35-58.