A second-order, unconditionally positive, mass-conserving integration scheme for biochemical systems

Abstract

C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a MAS Modelling, Analysis and Simulation Modelling, Analysis and SimulationA second-order, unconditionally positive, mass-conserving integration scheme for biochemical systems J. Bruggeman, H. Burchard, B. Kooi, B.P. Sommeijer A second-order, unconditionally positive, massconserving integration scheme for biochemical systems ABSTRACT Biochemical systems are bound by two mathematically-relevant restrictions. First, state variables in such systems represent non-negative quantities, such as concentrations of chemical compounds. Second, biochemical systems conserve mass and energy. Both properties must be reflected in results of an integration scheme applied to biochemical models. This paper first presents a mathematical framework for biochemical problems, which includes an exact definition of biochemical conservation: elements and energy, rather than state variable units, are conserved. We then analyze various fixed-step integration schemes, including traditional Euler-based schemes and the recently published modified Patankar schemes, and conclude that none of these deliver unconditional positivity and biochemical conservation in combination with higher-order accuracy. Finally, we present two new fixed-step integration schemes, one first-order and one second-order accurate, which do guarantee positivity and (biochemical) conservation. REPORT MAS-R0601 JANUARY 2006 AbstractBiochemical systems are bound by two mathematically-relevant restrictions. First, state variables in such systems represent non-negative quantities, such as concentrations of chemical compounds. Second, biochemical systems conserve mass and energy. Both properties must be reflected in results of an integration scheme applied to biochemical models. This paper first presents a mathematical framework for biochemical problems, which includes an exact definition of biochemical conservation: elements and energy, rather than state variable units, are conserved. We then analyze various fixed-step integration schemes, including traditional Euler-based schemes and the recently published Modified Patankar schemes, and conclude that none of these deliver unconditional positivity and biochemical conservation in combination with higher-order accuracy. Finally, we present two new fixed-step integration schemes, one first-order and one second-order accurate, which do guarantee positivity and (biochemical) conservation.

Iron profiles and speciation of the upper water column at the Bermuda Atlantic time-series Study site: a model based sensitivity study

“…Furthermore, GOTM provides stiff equation solvers appropriate for dealing with the very fast photochemical reaction rates in the model. In particular we used the recently developed non-negative and conservative modified Patankar-type solvers for ordinary differential equations (Burchard et al, 2005;Bruggeman et al, 2006). Meteorological data are derived from the ERA-40 reanalysis project (Uppala et al, 2005) to drive the physical model.…”

Combinations of nonstandard finite difference schemes and composition methods with complex time steps for population models

“…Using the strategy of adaptive time stepping, many schemes can achieve conditional positivity. In fluid flow dynamics, Courant-type conditions are used to ensure positivity, as well as stability [3].…”

Iron profiles and speciation of the upper water column at the Bermuda Atlantic Time-series Study site: a model based sensitivity study

“…Furthermore, GOTM provides stiff equation solvers appropriate for dealing with the very fast photochemical reaction rates in the model. In particular we used the recently developed non-negative and conservative modified Patankar-type solvers for ordinary differential equations Bruggeman et al, 2006). Meteorological data are derived from the ERA-40 reanalysis project (Uppala et al, 2005) to drive the physical model.…”

“…The physical part of the model is the General Ocean Turbulence Model (GOTM: www.gotm.net and Burchard et al (2006)). We choose GOTM because it uses state-of-theart turbulence closure models, which is a precondition for properly simulating vertical profiles of reactive biogeochemical quantities.…”