Ocean Dynamics

, Volume 55, Issue 3–4, pp 326–337 | Cite as

Application of modified Patankar schemes to stiff biogeochemical models for the water column

  • Hans BurchardEmail author
  • Eric Deleersnijder
  • Andreas Meister
Original paper


In this paper, we apply recently developed positivity preserving and conservative Modified Patankar-type solvers for ordinary differential equations to a simple stiff biogeochemical model for the water column. The performance of this scheme is compared to schemes which are not unconditionally positivity preserving (the first-order Euler and the second- and fourth-order Runge–Kutta schemes) and to schemes which are not conservative (the first- and second-order Patankar schemes). The biogeochemical model chosen as a test ground is a standard nutrient–phytoplankton–zooplankton–detritus (NPZD) model, which has been made stiff by substantially decreasing the half saturation concentration for nutrients. For evaluating the stiffness of the biogeochemical model, so-called numerical time scales are defined which are obtained empirically by applying high-resolution numerical schemes. For all ODE solvers under investigation, the temporal error is analysed for a simple exponential decay law. The performance of all schemes is compared to a high-resolution high-order reference solution. As a result, the second-order modified Patankar–Runge–Kutta scheme gives a good agreement with the reference solution even for time steps 10 times longer than the shortest numerical time scale of the problem. Other schemes do either compute negative values for non-negative state variables (fully explicit schemes), violate conservation (the Patankar schemes) or show low accuracy (all first-order schemes).


Phytoplankton Explicit Scheme Euler Scheme Biogeochemical Model Photosynthetically Available Radiation 
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Substantial parts of this work were carried out during a visit of Hans Burchard in Louvain-la-Neuve, funded by the Institut d’ Astronomie et de Géophysique G. Lemaî tre of the Université Catholique de Louvain. Eric Deleersnijder is a Research Associate with the Belgian National Fund for Scientific Research (FNRS) and his contribution to the present study was made in the scope of the project “A second-generation model of the ocean system”, which is funded by the Communauté Française de Belgique (Actions de Recherche Concertée) (see We are grateful to Karsten Bolding (Baaring, Denmark) who has provided the computational framework for the biogeochemical calculations carried out here. Finally, we acknowledge the critical comments of two anonymous reviewers who helped us to substantially improve the manuscript.


  1. Abdulle A (2001) Fourth order Chebyshev methods with recurrence relations. SIAM J Sci Comp 23:2042–2055Google Scholar
  2. Abdulle A, Medovikov AA (2001) Second order Chebyshev methods based on orthogonal polynomials. Numer Math 90:1–18CrossRefGoogle Scholar
  3. Bolding K, Burchard H, Pohlmann T, Stips A (2002) Turbulent mixing in the Northern North Sea: a numerical model study. Cont Shelf Res 22:2707–2724CrossRefGoogle Scholar
  4. Burchard H, Deleersnijder E, Meister A (2003) A high-order conservative Patankar-type discretisation for stiff systems of production-destruction equations. Appl Numer Math 47:1–30CrossRefGoogle Scholar
  5. Burchard H, Bolding K, Kühn W, Meister A, Neumann T,Umlauf L (2005) Description of a flexible and extendable physical-biogeochemical model system for the water column. J Mar Sys (in press)Google Scholar
  6. Deleersnijder E, Beckers J-M, Campin J-M, El Mohajir M, Fichefet T, Luyten P (1997) Some mathematical problems associated with the development and use of marine models. In: Diaz JI (ed) The mathematics of models for climatology and environment, vol 48. of NATO ASI Series, Springer, Berlin Heidelberg New York, pp 41–86Google Scholar
  7. Fasham MJR, Ducklow HW, McKelvie SM (1990) A nitrogen-based model of plankton dynamics in the oceanic mixed layer. J Mar Res 48:591–639Google Scholar
  8. Fennel W, Neumann T (1996) The mesoscale variability of nutrients and plankton as seen in a coupled model. Dt Hydrogr Z 48:49–71CrossRefGoogle Scholar
  9. Hairer E, Wanner G (2004) Solving ordinary differential equations II, Series in computational mathematics 14, 3rd edn. Springer, Berlin Heidelberg New YorkGoogle Scholar
  10. Hairer E, Nørsett S, Wanner G (2000) Solving ordinary differential equations I, Series in computational mathematics 8, 2nd edn. Springer, Berlin Heidelberg New YorkGoogle Scholar
  11. Harrison WG, Harris L, Irwin BD (1996) The kinetics of nitrogen utilization in the oceanic mixed layer: nitrate and ammonium interactions at nanomolar concentrations. Limnol Oceanogr 41:16–32CrossRefGoogle Scholar
  12. Horváth Z (1998) Positivity of Runge–Kutta and diagonally split Runge–Kutta methods. Appl Numer Math 28:309–326CrossRefGoogle Scholar
  13. van der Houwen PJ (1996) The development of Runge–Kutta methods for parabolic differential equations. Appl Num Math 20:261–273CrossRefGoogle Scholar
  14. van der Houwen PJ, Sommeijer BP (1980) On the internal stability of explicit m-stage Runge–Kutta methods for large m-values. ZAMM 60:479–485CrossRefGoogle Scholar
  15. Hundsdorfer W, Verwer JG (2003) Numerical solution of time-dependent advection-diffusion-reaction equations0, vol 33 of Series in computational mathematics. Springer, Berlin Heidelberg New YorkGoogle Scholar
  16. Kondo J (1975) Air-sea bulk transfer coefficients in diabatic conditions. Bound Layer Meteor 9:91–112CrossRefGoogle Scholar
  17. Kühn W, Radach G (1997) A one-dimensional physical-biological model study of the pelagic nitrogen cycling during the spring bloom in the northern North Sea (FLEX’76). J Mar Res 55:687–734CrossRefGoogle Scholar
  18. Lebedev V (2000) Explicit difference schemes for solving stiff problems with a complex or separable spectrum. Comp Math Math Phys 40:1729–1740Google Scholar
  19. Leonard BP (1991) The ULTIMATE conservative difference scheme applied to unsteady one-dimensional advection. Comput Meth Appl Mech Eng 88:17–74CrossRefGoogle Scholar
  20. Medovikov AA (1998) High order explicit methods for stiff ordinary differential equations. BIT 38:372–390CrossRefGoogle Scholar
  21. Meister A (1998) Comparison of different Krylov subspace methods embedded in an implicit finite volume scheme for the computation of viscous and inviscid flow fields on unstructured grids. J Comput Phys 140:311–345CrossRefGoogle Scholar
  22. Oschlies A, Kähler P (2004) Biotic contribution to air-sea fluxes of CO2 and O2 and its relation to new production, export production, and net community production. Global Biogeochemical Cycles 18. GB1015, doi:10.1029/2003GB002094Google Scholar
  23. Patankar SV (1980) Numerical heat transfer and fluid flow. McGraw-Hill, New YorkGoogle Scholar
  24. Pietrzak J (1998) The use of TVD limiters for forward-in-time upstream-biased advection schemes in ocean modeling. Mon Weather Rev 126:812–830CrossRefGoogle Scholar
  25. Popova EE, Ryabchenko VA, Fasham MJR (2000) Biological pump and vertical mixing in the southern ocean: their impact on atmospheric CO2. Global Biogeochem Cycles 14:477–498CrossRefGoogle Scholar
  26. Robertson HH (1966) The solution of a set of reaction rate equations. In: Walsh J (ed) Numerical analysis, an introduction. Academic, New York, pp 178–182Google Scholar
  27. Sandu A, Verwer J, van Loon M, Carmichael G, Potra F, Dabdub D, Seinfeld J (1997) Benchmarking stiff ODE solvers for atmospheric chemistry problems I: Implicit versus explicit. Atmos Environ 31:3151–3166CrossRefGoogle Scholar
  28. Soetaert K, Herman PMJ, Middelburg JJ (1996) A model of early diagenetic processes from the shelf to abyssal depths. Geochim Cosmochim Acta 60:1019–1040CrossRefGoogle Scholar
  29. Verwer JG (1996) Explicit Runge–Kutta methods for parabolic partial differential equations. Appl Num Math 22:359–379CrossRefGoogle Scholar
  30. Weber L, Völker C, Schartau M, Wolf-Gladrow DA (2005) Modelling the speciation and biochemistry of iron at the Bermuda Atlantik Time-series Study site, Global Biogeochemical Cycles, 19. doi:10.1029/2004GB002340Google Scholar

Copyright information

© Springer-Verlag 2005

Authors and Affiliations

  • Hans Burchard
    • 1
    Email author
  • Eric Deleersnijder
    • 2
  • Andreas Meister
    • 3
  1. 1.Baltic Sea Research Institute WarnemündeRostock-WarnemündeGermany
  2. 2.G. Lemaî tre Institute of Astronomy and Geophysics (ASTR) and Centre for Systems Engineering and Applied Mechanics (CESAME)Université catholique de LouvainLouvain-la-NeuveBelgium
  3. 3.Faculty of Mathematics and InformaticsUniversity of KasselKasselGermany

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