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Annals of Operations Research

, Volume 258, Issue 1, pp 79–106 | Cite as

Structure preserving integration and model order reduction of skew-gradient reaction–diffusion systems

  • Bülent Karasözen
  • Tuğba Küçükseyhan
  • Murat Uzunca
Article

Abstract

Activator-inhibitor FitzHugh–Nagumo (FHN) equation is an example for reaction–diffusion equations with skew-gradient structure. We discretize the FHN equation using symmetric interior penalty discontinuous Galerkin (SIPG) method in space and average vector field (AVF) method in time. The AVF method is a geometric integrator, i.e. it preserves the energy of the Hamiltonian systems and energy dissipation of the gradient systems. In this work, we show that the fully discrete energy of the FHN equation satisfies the mini-maximizer property of the continuous energy for the skew-gradient systems. We present numerical results with traveling fronts and pulses for one dimensional, two coupled FHN equations and three coupled FHN equations with one activator and two inhibitors in skew-gradient form. Turing patterns are computed for fully discretized two dimensional FHN equation in the form of spots and labyrinths. Because the computation of the Turing patterns is time consuming for different parameters, we applied model order reduction with the proper orthogonal decomposition (POD). The nonlinear term in the reduced equations is computed using the discrete empirical interpolation (DEIM) with SIPG discretization. Due to the local nature of the discontinuous Galerkin method, the nonlinear terms can be computed more efficiently than for the continuous finite elements. The reduced solutions are very close to the fully discretized ones. The efficiency and accuracy of the POD and POD–DEIM reduced solutions are shown for the labyrinth-like patterns.

Keywords

FitzHugh–Nagumo equations Gradient systems Traveling fronts and pulses Turing patterns Energy preservation Discontinuous Galerkin Model order reduction Discrete empirical interpolation 

Mathematics Subject Classification

35K57 65M60 35B36 

Notes

Acknowledgments

The authors would like to thank the reviewer for the comments and suggestions that help improve the manuscript. This work has been supported by METU BAP-07-05-2015 009.

References

  1. Antil, H., Heinkenschloss, M., & Sorensen, C. D. (2014). Application of the discrete empirical interpolation method to reduced order modeling of nonlinear and parametric systems. In A. Quarteroni & G. Rozza (Eds.), Reduced order methods for modeling and computational reduction, MS & A—modeling, simulation and applications (Vol. 9, pp. 101–136). Berlin: Springer International Publishing.Google Scholar
  2. Arnold, D. N. (1982). An interior penalty finite element method with discontinuous elements. SIAM Journal on Numerical Analysis, 19, 724–760.CrossRefGoogle Scholar
  3. Barrault, M., Maday, Y., Nguyen, N. C., & Patera, A. T. (2004). An empirical interpolation method: Application to efficient reduced-basis discretization of partial differential equations. Comptes Rendus Mathematique, 339(9), 667–672. doi: 10.1016/j.crma.2004.08.006.CrossRefGoogle Scholar
  4. Celledoni, E., Grimm, V., McLachlan, R. I., McLaren, D. I., O’Neale, D. J., Owren, B., et al. (2012). Preserving energy resp. dissipation in numerical PDEs using the “Average Vector Field” method. Journal of Computational Physics, 231, 6770–6789.CrossRefGoogle Scholar
  5. Chaturantabut, S., & Sorensen, D. C. (2010). Nonlinear model reduction via discrete empirical interpolation. SIAM Journal of Scientific Computation, 32(5), 2737–2764.CrossRefGoogle Scholar
  6. Chen, C. N., & Hu, X. (2014). Stability analysis for standing pulse solutions to FitzHugh–Nagumo equations. Calculus of Variations and Partial Differential Equations, 49, 827–845. doi: 10.1007/s00526-013-0601-0.CrossRefGoogle Scholar
  7. Grepl, M. A. (2012). Model order reduction of parametrized nonlinear reaction-diffusion systems. Computers & Chemical Engineering, 43, 33–44. doi: 10.1016/j.compchemeng.2012.03.013.CrossRefGoogle Scholar
  8. Hairer, E., & Lubich, C. (2014). Energy-diminishing integration of gradient systems. IMA Journal of Numerical Analysis, 34(2), 452–461. doi: 10.1093/imanum/drt031.CrossRefGoogle Scholar
  9. van Heijster, P., & Sandstede, B. (2011). Planar radial spots in a three-component FitzHugh–Nagumo system. Journal of Nonlinear Science, 21(5), 705–745. doi: 10.1007/s00332-011-9098-x.CrossRefGoogle Scholar
  10. van Heijster, P., Doelman, A., & Kaper, T. J. (2008). Pulse dynamics in a three-component system: Stability and bifurcations. Physica D: Nonlinear Phenomena, 237(24), 3335–3368. doi: 10.1016/j.physd.2008.07.014.CrossRefGoogle Scholar
  11. Kunisch, K., & Volkwein, S. (2001). Galerkin proper orthogonal decomposition methods for parabolic problems. Numerische Mathematik, 90(1), 117–148. doi: 10.1007/s002110100282.CrossRefGoogle Scholar
  12. Marquez-Lago, T. T., & Padilla, P. (2014). A selection criterion for patterns in reactiondiffusion systems. Theoretical Biology and Medical Modelling, 11, 7. doi: 10.1186/1742-4682-11-7.CrossRefGoogle Scholar
  13. Or-Guil, M., Bode, M., Schenk, C. P., & Purwins, H. G. (1998). Spot bifurcations in three-component reaction-diffusion systems: The onset of propagation. Physical Review E, 57, 6432–6437. doi: 10.1103/PhysRevE.57.6432.CrossRefGoogle Scholar
  14. Rivière, B. (2008). Discontinuous Galerkin methods for solving elliptic and parabolic equations. Society for Industrial and Applied Mathematics. doi: 10.1137/1.9780898717440.
  15. Tiso, P., & Rixen, D. J. (2013). Discrete empirical interpolation method for finite element structural dynamics. In: Topics in Nonlinear Dynamics, Volume 1 Proceedings of the 31st IMAC, A Conference on Structural Dynamics, Topics in nonlinear dynamics, Vol. 1, The Society for Experimental Mechanics, pp. 203–212.Google Scholar
  16. Turing, A. M. (1952). The chemical basis of morphogenesis. Philosophical Transactions of the Royal Society of London Series B, Biological Sciences, 237(641), 37–72. doi: 10.1098/rstb.1952.0012.CrossRefGoogle Scholar
  17. Yanagida, E. (2002a). Mini-maximizers for reaction-diffusion systems with skew-gradient structure. Journal of Differential Equations, 179, 311–335. doi: 10.1006/jdeq.2001.4028.CrossRefGoogle Scholar
  18. Yanagida, E. (2002b). Standing pulse solutions in reaction-diffusion systems with skew-gradient structure. Journal of Dynamics and Differential Equations, 14, 189–205. doi: 10.1023/A:1012915411490.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Bülent Karasözen
    • 1
  • Tuğba Küçükseyhan
    • 1
  • Murat Uzunca
    • 2
  1. 1.Department of Mathematics and Institute of Applied MathematicsMiddle East Technical UniversityAnkaraTurkey
  2. 2.Institute of Applied MathematicsMiddle East Technical UniversityAnkaraTurkey

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