Application of modified Patankar schemes to stiff biogeochemical models for the water column
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).
KeywordsPhytoplankton Explicit Scheme Euler Scheme Biogeochemical Model Photosynthetically Available Radiation
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 http://www.astr.ucl.ac.be/SLIM). 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.
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