Extension of modified Patankar-Runge-Kutta schemes to nonautonomous production-destruction systems based on Oliver’s approach. (English) Zbl 1459.65110

Summary: The mathematical modeling of various real life applications leads to systems of ordinary differential equations which include crucial properties like the positivity of the solution as well as the conservation of mass or energy. Based on the fundamental work of H. Burchard et al. [Appl. Numer. Math. 47, No. 1, 1–30 (2003; Zbl 1028.80008)], unconditionally positive and conservative modified Patankar-Runge-Kutta schemes (MPRK) are available. These methods are highly stable and often outperform standard Runge-Kutta schemes.
In this article, we extend MPRK methods, named MPRKO methods, using Oliver’s approach [J. Oliver, Math. Comput. 29, 1032–1036 (1975; Zbl 0331.65044)] to improve the accuracy of these schemes in the field of nonautonomous systems. The approach does not demand \(\mathbf{A} \mathbf{e} = \mathbf{c}\) in the Butcher tableau \(( \mathbf{A} , \mathbf{b} , \mathbf{c} )\), where \(\mathbf{e} = ( 1 , \dots , 1 )^T\). Following the general analysis of MPRK schemes described in [S. Kopecz and A. Meister, BIT 58, No. 3, 691–728 (2018; Zbl 1397.65102)], positivity and mass conservation fundamental properties are proven and even conditions concerning the Patankar weights are given to get second order accuracy of the MPRKO methods. Finally, we consider different linear models and a non-linear epidemiological SEIR problem to confirm the theoretical results and to give reliable statements about the accuracy of the novel class of MPRKO methods.


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] Bruggeman, J.; Burchard, H.; Kool, B. W.; Sommelier, B., A second-order, unconditionally positive, mass conserving integration scheme for biochemical systems, Appl. Numer. Math., 57, 36-58 (2007) · Zbl 1123.65067
[2] Kopecz, S.; Meister, A., A comparison of numerical methods for conservative and positive advection-diffusion-production-destruction systems, Proc. Appl. Math. Mech., 19, Article e201900209 pp. (2019)
[3] Sandu, A., Positive numerical integration methods for chemical kinetic systems, J. Comput. Phys., 170, 589-602 (2001) · Zbl 0984.65070
[4] Shampine, L. F.; Thompson, S.; Kierzenka, J. A.; Byrne, G. D., Non-negative solutions of ODEs, Appl. Math. Comput., 170, 556-569 (2005) · Zbl 1082.65547
[5] 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
[6] Shampine, L. F., Conservation laws and the numerical solution of ODEs, Comput. Math. Appl., 12, 1287-1296 (1986) · Zbl 0641.65057
[7] Butcher, J. C., Numerical Methods for Ordinary Differential Equations (2016), Wiley · Zbl 1354.65004
[8] Hairer, E.; Norsett, S. P.; Wanner, G., (Solving Ordinary Differential Equations I. Nonstiff Problems. Solving Ordinary Differential Equations I. Nonstiff Problems, Springer Series in Comp. Math., vol. 8 (2000), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0789.65048
[9] Hairer, E.; Wanner, G., (Solving Ordinary Differential Equations II. Stiff Problems. Solving Ordinary Differential Equations II. Stiff Problems, Springer Series in Comp. Math., vol. 14 (2004), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0859.65067
[10] Oliver, J., A curiosity of low-order explicit Runge-Kutta methods, Math. Comput., 29, 1032-1036 (1975) · Zbl 0331.65044
[11] Tsitouras, Ch., Explicit Runge-Kutta methods for starting integration of Lane-Emden problem, Appl. Math. Comput., 354, 353-364 (2019) · Zbl 1429.65149
[12] Bolley, C.; Crouzeix, M., Conservation de la positivité lors de la discrétisation des problèmes d’évolution paraboliques, ESAIM Math. Model. Numer. Anal., 12, 237-245 (1978) · Zbl 0392.65042
[13] Hundsdorfer, W.; Verwer, J., Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations (2003), Springer · Zbl 1030.65100
[14] Bertolazzi, E., Positive and conservative schemes for mass action kinetics, Comput. Math. Appl., 32, 29-43 (1996) · Zbl 0859.92030
[15] Hórvath, Z., Positivity of Runge-Kutta and diagonally split Runge-Kutta methods, Appl. Numer. Math., 28, 309-326 (1998) · Zbl 0926.65073
[16] Hórvath, Z., On the positivity step size threshold of Runge-Kutta methods, Appl. Numer. Math., 53, 341-356 (2005) · Zbl 1073.65077
[17] Gottlieb, S.; Ketcheson, D. I.; Shu, C.-W., Strong Stability Preserving Runge-Kutta and Multistep Time Discretizations (2011), World Scientific · Zbl 1241.65064
[18] Hundsdorfer, W.; Verwer, J.; Blom, J., Numerical time integration for air pollution models, Surv. Math. Ind., 10, 107-174 (2002) · Zbl 0999.65097
[19] Sandu, A., (Time-Stepping Methods that Favor Positivity for Atmospheric Chemistry Modeling. Time-Stepping Methods that Favor Positivity for Atmospheric Chemistry Modeling, IMA Math. Appl., vol. 130 (2002), Springer), 21-37 · Zbl 1108.80331
[20] Huang, J.; Shu, C. W., Positivity-preserving time discretizations for production-destruction equations with applications to non-equilibrium flows, J. Sci. Comput., 78, 1811-1839 (2018) · Zbl 1420.35190
[21] Huang, J.; Zhao, W.; Shu, C. W., A third-order unconditionally positivity-preserving scheme for production-destruction equations with applications to non-equilibrium flows, J. Sci. Comput., 79, 1015-1056 (2019) · Zbl 1444.35125
[22] Kopecz, S.; Meister, A., On order conditions for modified Patankar Runge-Kutta schemes, Appl. Numer. Math., 123, 159-179 (2018) · Zbl 1377.65089
[23] Kopecz, S.; Meister, A., Unconditional positive and conservative third order modified Patankar Runge Kutta discretizations of production destruction systems, BIT, 58, 694-728 (2018) · Zbl 1397.65102
[24] Kopecz, S.; Meister, A., On the existence of three-stage third-order modified Patankar-Runge-Kutta schemes, Numer. Algorithms, 81, 1473-1484 (2019) · Zbl 1416.65205
[25] Öffner, P.; Torlo, D., Arbitrary high-order conservative and positivity preserving Patankar-type deferred correction schemes, Appl. Numer. Math., 153, 15-34 (2020) · Zbl 1437.65073
[26] Patankar, S. V., Numerical Heat Transfer and Fluid Flow (1980), McGraw-Hill: McGraw-Hill New York · Zbl 0521.76003
[27] 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)
[28] 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. Mar. Syst., 61, 180-211 (2006)
[29] Gressel, O., Toward realistic simulations of magneto-thermal winds from weakly-ionized protoplanetary disks, J. Phys. Conf. Ser., 837, Article 012008 pp. (2017)
[30] Hense, I.; Beckmann, A., The representation of cyanobacteria life cycle processes in aquatic ecosystem models, Ecol. Model., 221, 2330-2338 (2010)
[31] Hense, I.; Burchard, H., Modelling cyanobacteria in shallow coastal seas, Ecol. Model., 221, 238-244 (2010)
[32] Klar, J. S.; Mücket, J. P., A detailed view of filaments and sheets in the warm-hot intergalactic medium, Astronom. Astrophys., 522 (2010), A114 · Zbl 1260.85136
[33] Semeniuk, K.; Dastoor, A., Development of a global ocean mercury model with a methylation cycle: outstanding issues, Glob. Biogeochem. Cycles, 31, 400-433 (2017)
[34] 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)
[35] Zill, D.; Cullen, M., Differential Equations with Boundary-Value Problems (2012), Brooks Cole Publishing Company. An International Thomson Publishing Company
[36] Friedmann, E.; Neumann, F.; Rannacher, R., Well-posedness of a linear spatio-temporal model of the JAK2/STAT5 signaling pathway, Commun. Math. Anal., 15, 76-102 (2013) · Zbl 1277.35210
[37] De la Sen, M.; Alonso-Quesada, S., Vaccination strategies based on feedback control techniques for a general SEIR-epidemic model, Appl. Math. Comput., 208, 3888-3904 (2011) · Zbl 1238.92030
[38] Shampine, L. F.; Reichelt, M. W., The MATLAB ODE suite, SIAM J. Sci. Comput., 18, 1-22 (1997) · Zbl 0868.65040
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