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Unconditionally positive and conservative third order modified Patankar-Runge-Kutta discretizations of production-destruction systems. (English) Zbl 1397.65102
Summary: Modified Patankar-Runge-Kutta (MPRK) schemes are numerical methods for the solution of positive and conservative production-destruction systems. They adapt explicit Runge-Kutta schemes to ensure positivity and conservation irrespective of the time step size. The first two members of this class, the first order MPE scheme and the second order MPRK22(1) scheme, have been successfully applied in a large number of applications. Recently, a general definition of MPRK schemes was introduced and necessary as well as sufficient conditions to obtain first and second order MPRK schemes were presented. In this paper we derive necessary and sufficient conditions for third order MPRK schemes and introduce the first family of such schemes. The theoretical results are confirmed by numerical experiments considering linear and nonlinear as well as nonstiff and stiff systems of differential equations.

MSC:
65L05 Numerical methods for initial value problems involving ordinary differential equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
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[1] Benz, J.; Meister, A.; Zardo, PA; Tadmor, E. (ed.); Liu, J-G (ed.); Tzavaras, A. (ed.), A conservative, positivity preserving scheme for advection-diffusion-reaction equations in biochemical applications, 399-408, (2009), Providence · Zbl 1189.35242
[2] Bonaventura, L.; Rocca, AD, Unconditionally strong stability preserving extensions of the TR-BDF2 method, J. Sci. Comput., 70, 859-895, (2017) · Zbl 1361.65046
[3] Broekhuizen, N.; Rickard, GJ; Bruggeman, J.; Meister, A., An improved and generalized second order, unconditionally positive, mass conserving integration scheme for biochemical systems, Appl. Numer. Math., 58, 319-340, (2008) · Zbl 1136.65067
[4] Bruggeman, J.; Burchard, H.; Kooi, BW; Sommeijer, B., A second-order, unconditionally positive, mass-conserving integration scheme for biochemical systems, Appl. Numer. Math., 57, 36-58, (2007) · Zbl 1123.65067
[5] Burchard, H.; Bolding, K.; Kühn, W.; Meister, A.; Neumann, T.; Umlauf, L., Description of a flexible and extendable physical-biogeochemical model system for the water column, J. Marine Syst., 61, 180-211, (2006)
[6] Burchard, H.; Deleersnijder, E.; Meister, A., A high-order conservative patankar-type discretisation for stiff systems of production-destruction equations, Appl. Numer. Math., 47, 1-30, (2003) · Zbl 1028.80008
[7] Burchard, H.; Deleersnijder, E.; Meister, A., Application of modified patankar schemes to stiff biogeochemical models for the water column, Ocean Dyn., 55, 326-337, (2005)
[8] Formaggia, L.; Scotti, A., Positivity and conservation properties of some integration schemes for mass action kinetics, SIAM J. Numer. Anal., 49, 1267-1288, (2011) · Zbl 1229.80020
[9] Gottlieb, S., Ketcheson, D., Shu, C.-W.: Strong Stability Preserving Runge-Kutta and Multistep Time Discretizations. World Scientific Publishing Co., Inc., Singapore (2011) · Zbl 1241.65064
[10] Gottlieb, S.; Shu, C-W, Total variation diminishing Runge-Kutta schemes, Math. Comput., 67, 73-85, (1998) · Zbl 0897.65058
[11] Gressel, O.: Toward realistic simulations of magneto-thermal winds from weakly-ionized protoplanetary disks. J. Phys. Conf. Ser. 837(1) (2017). https://doi.org/10.1088/1742-6596/837/1/012008
[12] Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations. I. Nonstiff Problems, 2nd edn. Springer, Berlin (1993) · Zbl 0789.65048
[13] Hense, I.; Beckmann, A., The representation of cyanobacteria life cycle processes in aquatic ecosystem models, Ecol. Model., 221, 2330-2338, (2010)
[14] Hense, I.; Burchard, H., Modelling cyanobacteria in shallow coastal seas, Ecol. Model., 221, 238-244, (2010)
[15] Horváth, Z., Positivity of Runge-Kutta and diagonally split Runge-Kutta methods, Appl. Numer. Math., 28, 309-326, (1998) · Zbl 0926.65073
[16] Howie, J.M.: Real Analysis, Springer Undergraduate Mathematics Series. Springer, London (2001)
[17] Klar, JS; Mücket, JP, A detailed view of filaments and sheets in the warm-hot intergalactic medium, Astronom. Astrophys., 522, a114, (2010) · Zbl 1260.85136
[18] Kopecz, S.; Meister, A., On order conditions for modified patankar-Runge-Kutta schemes, Appl. Numer. Math., 123, 159-179, (2018) · Zbl 1377.65089
[19] Lefever, R.; Nicolis, G., Chemical instabilities and sustained oscillations, J. Theor. Biol., 30, 267-284, (1971) · Zbl 1170.92344
[20] Meister, A., Benz, J.: Phosphorus Cycles in Lakes and Rivers: Modeling, Analysis, and Simulation, pp. 713-738. Springer, Berlin (2010) · Zbl 1197.86029
[21] Meister, A.; Butcher, JC, Sensitivity of modified patankar-type schemes for systems of conservative production-destruction equations, AIP Conf. Proc., 1863, 320006, (2017)
[22] Meister, A.; Ortleb, S., On unconditionally positive implicit time integration for the DG scheme applied to shallow water flows, Int. J. Numer. Methods Fluids, 76, 69-94, (2014)
[23] Ortleb, S., Hundsdorfer, W.: Patankar-type Runge-Kutta schemes for linear PDEs. AIP Conf. Proc. 1863(1) (2017). https://doi.org/10.1063/1.4992489
[24] Patankar, S.V.: Numerical Heat Transfer and Fluid Flow. Series in Computational Methods in Mechanics and Thermal Sciences. Hemisphere Pub. Corp., New York (1980)
[25] Radtke, H.; Burchard, H., A positive and multi-element conserving time stepping scheme for biogeochemical processes in marine ecosystem models, Ocean Model., 85, 32-41, (2015)
[26] Ralston, A., Rabinowitz, P.: A First Course in Numerical Analysis, 2nd edn. Dover Publications Inc., Mineola (2001) · Zbl 0976.65001
[27] Schippmann, B.; Burchard, H., Rosenbrock methods in biogeochemical modelling—a comparison to Runge-Kutta methods and modified patankar schemes, Ocean Model., 37, 112-121, (2011)
[28] Semeniuk, K.; Dastoor, A., Development of a global Ocean Mercury model with a methylation cycle: outstanding issues, Glob. Biogeochem. Cycles, 31, 400-433, (2017)
[29] Shu, C-W; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77, 439-471, (1988) · Zbl 0653.65072
[30] Strehmel, K., Weiner, R., Podhaisky, H.: Numerik gewöhnlicher Differentialgleichungen: Nichtsteife, steife und differential-algebraische Gleichungen. Springer, Berlin (2012) · Zbl 1256.65073
[31] Warns, A.; Hense, I.; Kremp, A., Modelling the life cycle of dinoflagellates: a case study with biecheleria baltica, J. Plankton Res., 35, 379-392, (2013)
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