Skip to main content
SpringerLink
Log in

Structure-Preserving Reduced- Order Modeling of Non-Traditional Shallow Water Equation

  • Chapter
  • First Online:
Model Reduction of Complex Dynamical Systems

Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 171))

  • 1269 Accesses

Abstract

An energy- preserving reduced -order model (ROM) is developed for the non-traditional shallow water equation (NTSWE) with full Coriolis force. The NTSWE in the noncanonical Hamiltonian/Poisson form is discretized in space by finite differences. The resulting system of ordinary differential equations is integrated in time by the energy preserving average vector field (AVF) method. The Poisson structure of the discretized NTSWE exhibits a skew-symmetric matrix depending on the state variables. An energy- preserving, computationally efficient reduced order model (ROM) is constructed by proper orthogonal decomposition with Galerkin projection. The nonlinearities are computed for the ROM efficiently by discrete empirical interpolation method. Preservation of the discrete energy and the discrete enstrophy are shown for the full- order model, and for the ROM which ensures the long- term stability of the solutions. The accuracy and computational efficiency of the ROMs are shown by two numerical test problems.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 119.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 159.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 159.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Afkham, B.M., Hesthaven, J.S.: Structure preserving model reduction of parametric Hamiltonian systems. SIAM J. Sci. Comput. 39(6), A2616–A2644 (2017). https://doi.org/10.1137/17M1111991

    Article  MathSciNet  MATH  Google Scholar 

  2. Astrid, P., Weiland, S., Willcox, K., Backx, T.: Missing point estimation in models described by proper orthogonal decomposition. IEEE Trans. Autom. Control 53(10), 2237–2251 (2008). https://doi.org/10.1109/TAC.2008.2006102

    Article  MathSciNet  MATH  Google Scholar 

  3. Barrault, M., Maday, Y., Nguyen, N.C., Patera, A.T.: An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. C. R. Math. 339(9), 667–672 (2004). https://doi.org/10.1016/j.crma.2004.08.006

    Article  MathSciNet  MATH  Google Scholar 

  4. Bauer, W., Cotter, C.: Energy-enstrophy conserving compatible finite element schemes for the rotating shallow water equations with slip boundary conditions. J. Comput. Phys. 373, 171–187 (2018). https://doi.org/10.1016/j.jcp.2018.06.071

    Article  MathSciNet  MATH  Google Scholar 

  5. Belanger, E., Vincent, A.: Data assimilation (4D-VAR) to forecast flood in shallow-waters with sediment erosion. J. Hydrol. 300(1–4), 114–125 (2005)

    Article  Google Scholar 

  6. Bistrian, D.A., Navon, I.M.: An improved algorithm for the shallow water equations model reduction: dynamic mode decomposition vs POD. Int. J. Numer. Methods Fluids 78(9), 552–580 (2015). https://doi.org/10.1002/fld.4029

    Article  MathSciNet  Google Scholar 

  7. Bistrian, D.A., Navon, I.M.: The method of dynamic mode decomposition in shallow water and a swirling flow problem. Int. J. Numer. Methods Fluids 83(1), 73–89 (2017)

    Article  MathSciNet  Google Scholar 

  8. Boss, E., Paldor, N., Thompson, L.: Stability of a potential vorticity front: from quasi-geostrophy to shallow water. J. Fluid Mech. 315, 65–84 (1996)

    Article  Google Scholar 

  9. Carlberg, K., Farhat, C., Cortial, J., Amsallem, D.: The GNAT method for nonlinear model reduction: effective implementation and application to computational fluid dynamics and turbulent flows. J. Comput. Phys. 242, 623–647 (2013). https://doi.org/10.1016/j.jcp.2013.02.028

    Article  MathSciNet  MATH  Google Scholar 

  10. Carlberg, K., Tuminaro, R., Boggs, P.: Preserving Lagrangian structure in nonlinear model reduction with application to structural dynamics. SIAM J. Sci. Comput. 37(2), B153–B184 (2015). https://doi.org/10.1137/140959602

    Article  MathSciNet  MATH  Google Scholar 

  11. Chaturantabut, S., Sorensen, D.C.: Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32(5), 2737–2764 (2010). https://doi.org/10.1137/090766498

    Article  MathSciNet  MATH  Google Scholar 

  12. Chaturantabut, S., Sorensen, D.C.: A state space error estimate for POD-DEIM nonlinear model reduction. SIAM J. Numer. Anal. 50(1), 46–63 (2012). https://doi.org/10.1137/110822724

    Article  MathSciNet  MATH  Google Scholar 

  13. Chaturantabut, S., Beattie, C., Gugercin, S.: Structure-preserving model reduction for nonlinear port-Hamiltonian systems. SIAM J. Sci. Comput. 38(5), B837–B865 (2016). https://doi.org/10.1137/15M1055085

    Article  MathSciNet  MATH  Google Scholar 

  14. Chertock, A., Dudzinski, M., Kurganov, A., Lukáčová-Medvid’ová, M.: Well-balanced schemes for the shallow water equations with Coriolis forces. Numer. Math. 138(4), 939–973 (2018). https://doi.org/10.1007/s00211-017-0928-0

  15. Cohen, D., Hairer, E.: Linear energy-preserving integrators for Poisson systems. BIT Numer. Math. 51(1), 91–101 (2011). https://doi.org/10.1007/s10543-011-0310-z

    Article  MathSciNet  MATH  Google Scholar 

  16. Cotter, C.J., Shipton, J.: Mixed finite elements for numerical weather prediction. J. Comput. Phys. 231(21), 7076–7091 (2012)

    Article  MathSciNet  Google Scholar 

  17. Dellar, P.J., Salmon, R.: Shallow water equations with a complete Coriolis force and topography. Phys. Fluids 17(10), 106601 (2005). https://doi.org/10.1063/1.2116747

    Article  MathSciNet  MATH  Google Scholar 

  18. Eldred, C., Dubos, T., Kritsikis, E.: A quasi-Hamiltonian discretization of the thermal shallow water equations. J. Comput. Phys. 379, 1–31 (2019). https://doi.org/10.1016/j.jcp.2018.10.038

    Article  MathSciNet  Google Scholar 

  19. Esfahanian, V., Ashrafi, K.: Equation-free/Galerkin-free reduced-order modeling of the shallow water equations based on Proper Orthogonal Decomposition. J. Fluids Eng. 131(7), 071401–13 (2009). https://doi.org/10.1115/1.3153368

    Article  Google Scholar 

  20. Fjordholm, U.S., Mishra, S., Tadmor, E.: Well-balanced and energy stable schemes for the shallow water equations with discontinuous topography. J. Comput. Phys. 230(14), 5587–5609 (2011). https://doi.org/10.1016/j.jcp.2011.03.042

    Article  MathSciNet  MATH  Google Scholar 

  21. Gassner, G.J., Winters, A.R., Kopriva, D.A.: A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water equations. Appl. Math. Comput. 272, 291–308 (2016). https://doi.org/10.1016/j.amc.2015.07.014

    Article  MathSciNet  MATH  Google Scholar 

  22. Gerkema, T., Shrira, V.I.: Near-inertial waves in the ocean: beyond the ‘traditional approximation’. J. Fluid Mech. 529, 195–219 (2005)

    Article  MathSciNet  Google Scholar 

  23. Gong, Y., Wang, Q., Wang, Z.: Structure-preserving Galerkin POD reduced-order modeling of Hamiltonian systems. Comput. Methods Appl. Mech. Eng. 315, 780–798 (2017). https://doi.org/10.1016/j.cma.2016.11.016

    Article  MathSciNet  MATH  Google Scholar 

  24. Karasözen, B., Uzunca, M.: Energy preserving model order reduction of the nonlinear Schrödinger equation. Adv. Comput. Math. 44(6), 1769–1796 (2018). https://doi.org/10.1007/s10444-018-9593-9

    Article  MathSciNet  MATH  Google Scholar 

  25. Lall, S., Krysl, P., Marsden, J.E.: Structure-preserving model reduction for mechanical systems. Phys. D 184(1–4), 304–318 (2003). https://doi.org/10.1016/S0167-2789(03)00227-6

    Article  MathSciNet  MATH  Google Scholar 

  26. Leibovich, S., Lele, S.: The influence of the horizontal component of Earth’s angular velocity on the instability of the Ekman layer. J. Fluid Mech. 150, 41–87 (1985)

    Article  Google Scholar 

  27. Lozovskiy, A., Farthing, M., Kees, C., Gildin, E.: POD-based model reduction for stabilized finite element approximations of shallow water flows. J. Comput. Appl. Math. 302, 50–70 (2016). https://doi.org/10.1016/j.cam.2016.01.029

    Article  MathSciNet  MATH  Google Scholar 

  28. Lozovskiy, A., Farthing, M., Kees, C.: Evaluation of Galerkin and Petrov-Galerkin model reduction for finite element approximations of the shallow water equations. Comput. Methods Appl. Mech. Eng. 318, 537–571 (2017). https://doi.org/10.1016/j.cma.2017.01.027

    Article  MathSciNet  MATH  Google Scholar 

  29. Lynch, P.: Hamiltonian methods for geophysical fluid dynamics: an introduction (2002)

    Google Scholar 

  30. Marshall, J., Schott, F.: Open-ocean convection: observations, theory, and models. Rev. Geophys. 37(1), 1–64 (1999)

    Article  Google Scholar 

  31. Miyatake, Y.: Structure-preserving model reduction for dynamical systems with a first integral. Jpn. J. Ind. Appl. Math. 36(3), 1021–1037 (2019). https://doi.org/10.1007/s13160-019-00378-y

    Article  MathSciNet  MATH  Google Scholar 

  32. Nguyen, N.C., Patera, A.T., Peraire, J.: A “best points” interpolation method for efficient approximation of parametrized functions. Int. J. Numer. Methods Eng. 73(4), 521–543 (2008). https://doi.org/10.1002/nme.2086

    Article  MathSciNet  MATH  Google Scholar 

  33. Ohlberger, M., Rave, S.: Reduced basis methods: success, limitations and future challenges. In: Proceedings of the Conference Algoritmy, pp. 1–12 (2016)

    Google Scholar 

  34. Peng, L., Mohseni, K.: Symplectic model reduction of Hamiltonian systems. SIAM J. Sci. Comput. 38(1), A1–A27 (2016). https://doi.org/10.1137/140978922

    Article  MathSciNet  MATH  Google Scholar 

  35. Ranocha, H.: Shallow water equations: split-form, entropy stable, well-balanced, and positivity preserving numerical methods. GEM Int. J. Geomath. 8(1), 85–133 (2017). https://doi.org/10.1007/s13137-016-0089-9

    Article  MathSciNet  MATH  Google Scholar 

  36. Salmon, R.: Hamiltonian fluid mechanics. Annu. Rev. Fluid Mech. 20(1), 225–256 (1988). https://doi.org/10.1146/annurev.fl.20.010188.001301

    Article  Google Scholar 

  37. Salmon, R.: Poisson-bracket approach to the construction of energy- and potential-enstrophy-conserving algorithms for the shallow-water equations. J. Atmos. Sci. 61(16), 2016–2036 (2004). https://doi.org/10.1175/1520-0469(2004)0612016:PATTCO2.0.CO;2

    Article  MathSciNet  Google Scholar 

  38. Ştefănescu, R., Navon, I.M.: POD/DEIM nonlinear model order reduction of an ADI implicit shallow water equations model. J. Comput. Phys. 237, 95–114 (2013). https://doi.org/10.1016/j.jcp.2012.11.035

    Article  MathSciNet  MATH  Google Scholar 

  39. Ştefănescu, R., Sandu, A., Navon, I.M.: Comparison of pod reduced order strategies for the nonlinear 2D shallow water equations. Int. J. Numer. Methods Fluids 76(8), 497–521 (2014). https://doi.org/10.1002/fld.3946

    Article  MathSciNet  Google Scholar 

  40. Stewart, A.L., Dellar, P.J.: Multilayer shallow water equations with complete Coriolis force. part 1. derivation on a non-traditional beta-plane. J. Fluid Mech. 651, 387-413 (2010). https://doi.org/10.1017/S0022112009993922

  41. Stewart, A.L., Dellar, P.J.: Multilayer shallow water equations with complete Coriolis force. part 3. hyperbolicity and stability under shear. J. Fluid Mech. 723, 289–317 (2013)

    Google Scholar 

  42. Stewart, A.L., Dellar, P.J.: An energy and potential enstrophy conserving numerical scheme for the multi-layer shallow water equations with complete Coriolis force. J. Comput. Phys. 313, 99–120 (2016). https://doi.org/10.1016/j.jcp.2015.12.042

    Article  MathSciNet  MATH  Google Scholar 

  43. Vallis, G.K.: Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press, Cambridge (2017)

    Book  Google Scholar 

  44. Warneford, E.S., Dellar, P.J.: Thermal shallow water models of geostrophic turbulence in Jovian atmospheres. Phys. Fluids 26(1), 016603 (2014)

    Article  Google Scholar 

  45. Wimmer, G., Cotter, C., Bauer, W.: Energy conserving upwinded compatible finite element schemes for the rotating shallow water equations. arXiv preprint (2019)

    Google Scholar 

  46. Xu, Y., van der Vegt, J.J.W., Bokhove, O.: Discontinuous Hamiltonian finite element method for linear hyperbolic systems. J. Sci. Comput. 35(2–3), 241–265 (2008). https://doi.org/10.1007/s10915-008-9191-y

    Article  MathSciNet  MATH  Google Scholar 

  47. Zimmermann, R., Willcox, K.: An accelerated greedy missing point estimation procedure. SIAM J. Sci. Comput. 38(5), A2827–A2850 (2016). https://doi.org/10.1137/15M1042899

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by 100/2000 Ph.D. Scholarship Program of the Turkish Higher Education Council. The authors would like to thank to the referees which helped to improve the paper.

Author information

Authors and Affiliations

  1. Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey

    Süleyman Yildiz

  2. Department of Mathematics, Sinop University, Sinop, Turkey

    Murat Uzunca

  3. Institute of Applied Mathematics & Department of Mathematics, Middle East Technical University, Ankara, Turkey

    Bülent Karasözen

Authors
  1. Süleyman Yildiz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Murat Uzunca
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bülent Karasözen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Süleyman Yildiz .

Editor information

Editors and Affiliations

  1. Dynamics of Complex Technical Systems, Max Planck Institute, Magdeburg, Sachsen-Anhalt, Germany

    Peter Benner

  2. Institute of Mathematics, Technical University of Berlin, Berlin, Germany

    Tobias Breiten

  3. Institute for Numerical Analysis, TU Braunschweig, Braunschweig, Germany

    Heike Faßbender

  4. Mathematisches Institut, Universität Koblenz-Landau, Campus Koblenz, Koblenz, Germany

    Michael Hinze

  5. Institut für Mathematik, Universität Augsburg, Augsburg, Germany

    Tatjana Stykel

  6. Department of Mathematics and Computer Science, University of Southern Denmark, Odense, Denmark

    Ralf Zimmermann

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Yildiz, S., Uzunca, M., Karasözen, B. (2021). Structure-Preserving Reduced- Order Modeling of Non-Traditional Shallow Water Equation. In: Benner, P., Breiten, T., Faßbender, H., Hinze, M., Stykel, T., Zimmermann, R. (eds) Model Reduction of Complex Dynamical Systems. International Series of Numerical Mathematics, vol 171. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-72983-7_15

Download citation

Publish with us

Policies and ethics

Search

Navigation