Issues with positivity-preserving Patankar-type schemes. (English) Zbl 1506.65106
Summary: Patankar-type schemes are linearly implicit time integration methods designed to be unconditionally positivity-preserving. However, there are only little results on their stability or robustness. We suggest two approaches to analyze the performance and robustness of these methods. In particular, we demonstrate problematic behaviors of these methods that, even on very simple linear problems, can lead to undesired oscillations and order reduction for vanishing initial condition. Finally, we demonstrate in numerical simulations that our theoretical results for linear problems apply analogously to nonlinear stiff problems.
MSC:
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
Keywords:
positivity preservation; Patankar-type methods; Runge-Kutta methods; deferred correction methods; implicit-explicit methods; semi-implicit methodsReferences:
| [1] | Abgrall, Rémi, High order schemes for hyperbolic problems using globally continuous approximation and avoiding mass matrices, J. Sci. Comput., 73, 2, 461-494 (2017) · Zbl 1398.65242 |
| [2] | Ávila, Andrés I.; Kopecz, Stefan; Meister, Andreas, A comprehensive theory on generalized BBKS schemes, Appl. Numer. Math., 157, 19-37 (2020) · Zbl 1452.80017 |
| [3] | Axelsson, Owe, Iterative Solution Methods (1996), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0845.65011 |
| [4] | Bellen, Alfredo; Torelli, Lucio, Unconditional contractivity in the maximum norm of diagonally split Runge-Kutta methods, SIAM J. Numer. Anal., 34, 2, 528-543 (1997) · Zbl 0873.65074 |
| [5] | Bezanson, Jeff; Edelman, Alan; Karpinski, Stefan; Shah, Viral B., Julia: a fresh approach to numerical computing, SIAM Rev., 59, 1, 65-98 (2017) · Zbl 1356.68030 |
| [6] | Bolley, Catherine; Crouzeix, Michel, Conservation de la positivité lors de la discrétisation des problèmes d’évolution paraboliques, RAIRO. Anal. Numér., 12, 3, 237-245 (1978) · Zbl 0392.65042 |
| [7] | Broekhuizen, N.; Rickard, G. J.; Bruggeman, J.; Meister, A., An improved and generalized second order, unconditionally positive, mass conserving integration scheme for biochemical systems, Appl. Numer. Math., 58, 3, 319-340 (2008) · Zbl 1136.65067 |
| [8] | Bruggeman, Jorn; Burchard, Hans; Kooi, Bob W.; Sommeijer, Ben, A second-order, unconditionally positive, mass-conserving integration scheme for biochemical systems, Appl. Numer. Math., 57, 1, 36-58 (2007) · Zbl 1123.65067 |
| [9] | Burchard, Hans; Deleersnijder, Eric; Meister, Andreas, 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 |
| [10] | Chertock, Alina; Cui, Shumo; Kurganov, Alexander; Wu, Tong, Steady state and sign preserving semi-implicit Runge-Kutta methods for ODEs with stiff damping term, SIAM J. Numer. Anal., 53, 4, 2008-2029 (2015) · Zbl 1327.65128 |
| [11] | Ciallella, Mirco; Micalizzi, Lorenzo; Öffner, Philipp; Torlo, Davide, An arbitrary high order and positivity preserving method for the shallow water equations (2021), arXiv preprint: |
| [12] | Fekete, Imre; Ketcheson, David I.; Lóczi, Lajos, Positivity for convective semi-discretizations, J. Sci. Comput., 74, 1, 244-266 (2018) · Zbl 1419.65043 |
| [13] | Formaggia, L.; Scotti, A., Positivity and conservation properties of some integration schemes for mass action kinetics, SIAM J. Numer. Anal., 49, 3, 1267-1288 (2011) · Zbl 1229.80020 |
| [14] | Frolkovic, Peter, Semi-implicit methods based on inflow implicit and outflow explicit time discretization of advection, (Proceedings of ALGORITMY (2016)), 165-174 |
| [15] | Gottlieb, Sigal; Ketcheson, David I.; Shu, Chi-Wang, Strong Stability Preserving Runge-Kutta and Multistep Time Discretizations (2011), World Scientific: World Scientific Singapore · Zbl 1241.65064 |
| [16] | Hairer, Ernst; Norsett, Syvert Paul; Wanner, Gerhard, (Hairer, E.; Norsett, S. P.; Wanner, G., Solving Ordinary, Differential Equations I, Nonstiff Problems, vol. 1. Solving Ordinary, Differential Equations I, Nonstiff Problems, vol. 1, 2000 (2000), Springer-Verlag) · Zbl 1185.65115 |
| [17] | Hairer, Ernst; Wanner, Gerhard, Stiff differential equations solved by Radau methods, J. Comput. Appl. Math., 111, 1-2, 93-111 (1999) · Zbl 0945.65080 |
| [18] | Horváth, Zoltán, Positivity of Runge-Kutta and diagonally split Runge-Kutta methods, Appl. Numer. Math., 28, 2-4, 309-326 (1998) · Zbl 0926.65073 |
| [19] | Huang, Juntao; Shu, Chi-Wang, Positivity-preserving time discretizations for production-destruction equations with applications to non-equilibrium flows, J. Sci. Comput., 78, 3, 1811-1839 (2019) · Zbl 1420.35190 |
| [20] | Huang, Juntao; Zhao, Weifeng; Shu, Chi-Wang, A third-order unconditionally positivity-preserving scheme for production-destruction equations with applications to non-equilibrium flows, J. Sci. Comput., 79, 2, 1015-1056 (2019) · Zbl 1444.35125 |
| [21] | in’t Hout, Karel J., A note on unconditional maximum norm contractivity of diagonally split Runge-Kutta methods, SIAM J. Numer. Anal., 33, 3, 1125-1134 (1996) · Zbl 0855.65095 |
| [22] | Izgin, Thomas; Kopecz, Stefan; Meister, Andreas, Recent developments in the field of modified Patankar-Runge-Kutta-methods, PAMM, 21, 1, Article e202100027 pp. (2021) |
| [23] | Izgin, Thomas; Kopecz, Stefan; Meister, Andreas, On Lyapunov stability of positive and conservative time integrators and application to second order modified Patankar-Runge-Kutta schemes (2022), arXiv preprint · Zbl 1492.65191 |
| [24] | Izgin, Thomas; Kopecz, Stefan; Meister, Andreas, On the stability of unconditionally positive and linear invariants preserving time integration schemes (2022), arXiv preprint · Zbl 1492.65191 |
| [25] | Kopecz, Stefan; Meister, Andreas, On order conditions for modified Patankar-Runge-Kutta schemes, Appl. Numer. Math., 123, 159-179 (2018) · Zbl 1377.65089 |
| [26] | Kopecz, Stefan; Meister, Andreas, Unconditionally positive and conservative third order modified Patankar-Runge-Kutta discretizations of production-destruction systems, BIT Numer. Math., 58, 3, 691-728 (2018) · Zbl 1397.65102 |
| [27] | Kopecz, Stefan; Meister, Andreas, A comparison of numerical methods for conservative and positive advection-diffusion-production-destruction systems, PAMM, 19, 1 (2019) · Zbl 1416.65205 |
| [28] | Kopecz, Stefan; Meister, Andreas, On the existence of three-stage third-order modified Patankar-Runge-Kutta schemes, Numer. Algorithms, 1-12 (2019) · Zbl 1416.65205 |
| [29] | Kuzmin, Dmitri, Entropy stabilization and property-preserving limiters for \(\mathbb{P}^1\) discontinuous Galerkin discretizations of scalar hyperbolic problems, J. Numer. Math., 29, 4, 307-322 (2021) · Zbl 1481.65228 |
| [30] | Macdonald, Colin B.; Gottlieb, Sigal; Ruuth, Steven J., A numerical study of diagonally split Runge-Kutta methods for PDEs with discontinuities, J. Sci. Comput., 36, 1, 89-112 (2008) · Zbl 1203.65167 |
| [31] | Martiradonna, Angela; Colonna, Gianpiero; Diele, Fasma, GeCo: geometric conservative nonstandard schemes for biochemical systems, Appl. Numer. Math., 155, 38-57 (2020) · Zbl 1436.65083 |
| [32] | Mazzia, Francesca; Magherini, Cecilia, Test set for initial value problem solvers (2008), Department of Mathematics, University of Bari: Department of Mathematics, University of Bari Italy, Technical Report Release 2.4 |
| [33] | Meister, Andreas; Ortleb, Sigrun, A positivity preserving and well-balanced dg scheme using finite volume subcells in almost dry regions, Appl. Math. Comput., 272, 259-273 (2016) · Zbl 1410.76250 |
| [34] | Mikula, Karol; Ohlberger, Mario, Inflow-implicit/outflow-explicit scheme for solving advection equations, (Finite Volumes for Complex Applications VI Problems & Perspectives. Finite Volumes for Complex Applications VI Problems & Perspectives, Springer Proceedings in Mathematics, vol. 4 (2011), Springer: Springer Berlin, Heidelberg), 683-691 · Zbl 1246.65167 |
| [35] | Mikula, Karol; Ohlberger, Mario; Urbán, Jozef, Inflow-implicit/outflow-explicit finite volume methods for solving advection equations, Appl. Numer. Math., 85, 16-37 (2014) · Zbl 1310.65103 |
| [36] | Nüßlein, Stephan; Ranocha, Hendrik; Ketcheson, David I., Positivity-preserving adaptive Runge-Kutta methods, Commun. Appl. Math. Comput. Sci., 16, 2, 155-179 (2021) · Zbl 1480.65175 |
| [37] | Öffner, Philipp; Torlo, Davide, Arbitrary high-order, conservative and positivity preserving Patankar-type deferred correction schemes, Appl. Numer. Math., 153, 15-34 (2020) · Zbl 1437.65073 |
| [38] | Patankar, Suhas V., Numerical Heat Transfer and Fluid Flow (1980), Hemisphere Publishing Corporation: Hemisphere Publishing Corporation Washington · Zbl 0521.76003 |
| [39] | Pratt, Orson, New and Easy Method of Solution of the Cubic Biquadratic Equations: Embracing Several New Formulas, Greatly Simplifying This Department of Mathematical Science (1866), Longmans: Longmans Green, Reader, and Dyer, Liverpool |
| [40] | Rackauckas, Christopher; Nie, Qing, DifferentialEquations.jl – a performant and feature-rich ecosystem for solving differential equations in Julia, J. Open Res. Softw., 5, 1, 15 (2017) |
| [41] | Ranocha, Hendrik, On strong stability of explicit Runge-Kutta methods for nonlinear semibounded operators, IMA J. Numer. Anal., 41, 1, 654-682 (2021) · Zbl 1464.65075 |
| [42] | Ranocha, Hendrik; Ketcheson, David I., Energy stability of explicit Runge-Kutta methods for nonautonomous or nonlinear problems, SIAM J. Numer. Anal., 58, 6, 3382-3405 (2020) · Zbl 1457.65032 |
| [43] | Ranocha, Hendrik; Öffner, Philipp, \( L_2\) stability of explicit Runge-Kutta schemes, J. Sci. Comput., 75, 2, 1040-1056 (2018) · Zbl 1398.65188 |
| [44] | Sun, Zheng; Shu, Chi-Wang, Stability of the fourth order Runge-Kutta method for time-dependent partial differential equations, Ann. Math. Sci. Appl., 2, 2, 255-284 (2017) · Zbl 1381.65079 |
| [45] | Sun, Zheng; Shu, Chi-Wang, Strong stability of explicit Runge-Kutta time discretizations, SIAM J. Numer. Anal., 57, 3, 1158-1182 (2019) · Zbl 1422.65224 |
| [46] | Torlo, Davide; Öffner, Philipp; Ranocha, Hendrik, Issues with positivity preserving Patankar-type schemes (2021), Git repository: |
| [47] | Torlo, Davide; Öffner, Philipp; Ranocha, Hendrik, Issues with positivity-preserving Patankar-type schemes (2021), arXiv preprint: |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
