A second-order, unconditionally positive, mass-conserving integration scheme for biochemical systems
Bruggeman, J. and Burchard, H. and Kooi, B. W. and Sommeijer, B. 2007.
A second-order, unconditionally positive, mass-conserving integration
scheme for biochemical systems. Appl Num. Math. 57: 36 - 58
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.