Rosenbrock methods in biogeochemical modelling – A comparison to Runge–Kutta methods and modified Patankar schemes

Modelling biogeochemical processes in the surface ocean is still a difficult task due to the challenge to identify the most convenient integration scheme for the reaction terms. The scheme is expected to deal with the model characteristics of positivity and conservativity as well as with the different time scales involved, which occur e.g., whenever photochemical reactions take place in the water column.This paper presents a numerical comparison of the Rosenbrock methods, ROS3 and ROS4, often used for solving chemical reactions, to the explicit fourth-order Runge–Kutta method and the unconditionally positive modified Patankar schemes. Following their successful application in air chemistry, we here test the hypothesis that the Rosenbrock methods are an optimal choice for marine biogeochemical modelling in terms of efficiency and accuracy. In this study the schemes are compared in terms of runtime and accuracy and are applied to two test cases of different complexity: a zero-dimensional nutrient–phytoplankton–detritus (NPD)-type model and a one-dimensional nutrient–phytoplankton–zooplankton–detritus (NPZD)-type model. Applying the Rosenbrock methods to the simple NPD model shows their advantage over the other applied methods. They give the most accurate results of all solvers, especially for large step sizes, in less computing time due to their semi-implicitness and adaptive step sizing. On the contrary, for the one-dimensional NPZD model problem this is only the case in comparison to the Runge–Kutta solver, while their performance is worse than that of the second-order modified Patankar scheme. They need longer runtimes than the latter ones in order to achieve similarly accurate results. However, the modified Patankar schemes are not conservative if the system reactions contain more than one source compound. Thus, for more complex marine biogeochemical problems, it is recommended to apply the Rosenbrock methods while for simpler models the use of the second-order modified Patankar method is still the best alternative.

Research highlights► We applied the Runge–Kutta solver, the modified Patankar schemes and two Rosenbrock solvers to two test cases of different complexity: (1) a zero-dimensional NPD-type model, (2) a one-dimensional NPZD-type model ► The main focus was on computational efficiency ► Modified Patankar schemes are the best choice to solve biogeochemical models if there are no multiple source compounds per system reactions. ► For more complex biogeochemical models application of the RBMs is advisable. ► In case of complex ecosystem models the gain of efficiency compared to other applicable models will be higher than the possible drawbacks of the adaptive time stepping technique, especially in complex realistic applications.