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A second-order, unconditionally positive, mass-conserving integration scheme for biochemical systems. (English) Zbl 1123.65067
The authors consider the numerical simulation of biochemical systems which satisfy conservation and positivity laws. Even though standard numerical schemes do not maintain conservation and positivity, two new low order methods are introduced which do obey these restrictions.

65L05 Numerical methods for initial value problems involving ordinary differential equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
92C40 Biochemistry, molecular biology
Full Text: DOI
[1] Bolley, C.; Crouzeix, M., Conservation de la positivité lors de la discrétisation des problèmes d’évolution parabolic, RAIRO anal. numer., 12, 3, 237-245, (1978) · Zbl 0392.65042
[2] Burchard, H.; Bolding, K.; Kuhn, W.; Meister, A.; Neumann, T.; Umlauf, L., Description of a flexible and extendable physical – biogeochemical model system for the water column, J. marine systems, 61, 3-4, 180-211, (2006)
[3] Burchard, H.; Deleersnijder, E.; Meister, A., A high-order conservative patankar-type discretisation for stiff systems of production – destruction equations, Appl. numer. math., 47, 1, 1-30, (2003) · Zbl 1028.80008
[4] Burchard, H.; Deleersnijder, E.; Meister, A., Application of modified patankar schemes of stiff biogeochemical models for the water column, Ocean dynamics, 55, 3-4, 326-337, (2005)
[5] Fasham, M.J.R.; Ducklow, H.W.; Mckelvie, S.M., A nitrogen-based model of plankton dynamics in the oceanic mixed layer, J. marine res., 48, 3, 591-639, (1990)
[6] Hairer, E.; Nørsett, S.P.; Wanner, G., Solving ordinary differential equations I: nonstiff problems, Springer ser. comput. math., vol. 8, (1993), Springer Berlin · Zbl 0789.65048
[7] Hanegraaf, P.P.F.; Kooi, B.W., The dynamics of a tri-trophic food chain with two-component populations from a biochemical perspective, Ecological modelling, 152, 1, 47-64, (2002) · Zbl 1323.92175
[8] Hundsdorfer, W.H.; Verwer, J.G., Numerical solution of time-dependent advection – diffusion – reaction equations, Springer ser. comput. math., vol. 33, (2003), Springer Berlin · Zbl 1030.65100
[9] Jansen, H.; Twizell, E.H., An unconditionally convergent discretization of the SEIR model, Math. comput. simulation, 58, 2, 147-158, (2002) · Zbl 0983.92025
[10] Kooi, B.W.; Hanegraaf, P.P.F., Bi-trophic food chain dynamics with multiple component populations, Bull. math. biol., 63, 2, 271-299, (2001) · Zbl 1323.92175
[11] Kooijman, S.A.L.M., The synthesizing unit as model for the stoichiometric fusion and branching of metabolic fluxes, Biophys. chem., 73, 1-2, 179-188, (1998)
[12] Kooijman, S.A.L.M., Dynamic energy and mass budgets in biological systems, (2000), Cambridge University Press Cambridge · Zbl 1162.92008
[13] Kuijper, L.D.J.; Kooi, B.W.; Anderson, T.R.; Kooijman, S.A.L.M., Stoichiometry and food-chain dynamics, Theoret. population biol., 66, 4, 323-339, (2004) · Zbl 1073.92055
[14] Mickens, R.E., Applications of nonstandard finite difference schemes, (2000), World Scientific Singapore · Zbl 1237.76105
[15] Neumann, T.; Fennel, W.; Kremp, C., Experimental simulations with an ecosystem model of the baltic sea: A nutrient load reduction experiment, Global biogeochem. cycles, 16, 3, (2002)
[16] Patankar, S.V., Numerical heat transfer and fluid flow, Series in computational methods in mechanics and thermal sciences, (1980), McGraw-Hill New York · Zbl 0595.76001
[17] Pietrzak, J., The use of TVD limiters for forward-in-time upstream-biased advection schemes in Ocean modeling, Monthly weather rev., 126, 3, 812-830, (1998)
[18] Piyawong, W.; Twizell, E.H.; Gumel, A.B., An unconditionally convergent finite-difference scheme for the SIR model, Appl. math. comput., 146, 2-3, 611-625, (2003) · Zbl 1026.92041
[19] Reder, C., Metabolic control-theory—a structural approach, J. theoret. biol., 135, 2, 175-201, (1988)
[20] Roels, J.A., Energetics and kinetics in biotechnology, (1983), Elsevier Biomedical Press Amsterdam
[21] Sandu, A., Positive numerical integration methods for chemical kinetic systems, J. comput. phys., 170, 2, 589-602, (2001) · Zbl 0984.65070
[22] Shampine, L.F., Conservations laws and the numerical solution of odes, Comput. math. appl. B, 12, 5-6, 1287-1296, (1986) · Zbl 0641.65057
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