Open Access
Volume 56, Number 6, November-December 2022
Page(s) 1843 - 1870
Published online 12 August 2022
  1. A. Alvermann and H. Fehske, High-order commutator-free exponential time-propagation of driven quantum systems. J. Comput. Phys. 230 (2011) 5930–5956. [CrossRef] [MathSciNet] [Google Scholar]
  2. A.I. Ávila, S. Kopecz and A. Meister, A comprehensive theory on generalized BBKS schemes. Appl. Numer. Math. 157 (2020) 19–37. [CrossRef] [MathSciNet] [Google Scholar]
  3. M. Beck and M.J. Gander, On the positivity of Poisson integrators for the Lotka-Volterra equations. BIT Numer. Math. 55 (2015) 319–340. [CrossRef] [Google Scholar]
  4. A. Berman and R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences. Computer Science and Applied Mathematics. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London (1979). [Google Scholar]
  5. E. Bertolazzi, Positive and conservative schemes for mass action kinetics. Comput. Math. App. 32 (1996) 29–43. [Google Scholar]
  6. S. Blanes, On the construction of symmetric second order methods for ODEs. Appl. Math. Lett. 98 (2019) 41–48. [CrossRef] [MathSciNet] [Google Scholar]
  7. S. Blanes and F. Casas, A Concise Introduction to Geometric Numerical Integration. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL (2016). [Google Scholar]
  8. S. Blanes, F. Casas, J.A. Oteo and J. Ros, The Magnus expansion and some of its applications. Phys. Rep. 470 (2009) 151–238. [CrossRef] [MathSciNet] [Google Scholar]
  9. S. Blanes, F. Casas and M. Thalhammer, High-order commutator-free quasi-Magnus exponential integrators for non-autonomous linear evolution equations. Comput. Phys. Commun. 220 (2017) 243–262. [CrossRef] [MathSciNet] [Google Scholar]
  10. C. Bolley and M. Crouzeix, Conservation de la positivité lors de la discrétisation des problèmes d’évolution paraboliques. RAIRO: Anal. Numér. 12 (1978) 237–245. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  11. N. Broekhuizen, G.J. Rickard, J. Bruggeman and A. Meister, An improved and generalized second order, unconditionally positive, mass conserving integration scheme for biochemical systems. Appl. Numer. Math. 58 (2008) 319–340. [CrossRef] [MathSciNet] [Google Scholar]
  12. J. Bruggeman, H. Burchard, B.W. Kooi and B. Sommeijer, A second-order, unconditionally positive, mass-conserving integration scheme for biochemical systems. Appl. Numer. Math. 57 (2007) 36–58. [CrossRef] [MathSciNet] [Google Scholar]
  13. H. Burchard, E. Deleersnijder and A. Meister, Application of modified Patankar schemes to stiff biogeochemical models for the water column. Ocean Dyn. 55 (2005) 326–337. [NASA ADS] [CrossRef] [Google Scholar]
  14. H. Burchard, E. Deleersnijder and A. Meister, A high-order conservative Patankar-type discretisation for stiff systems of production-destruction equations. Appl. Numer. Math. 47 (2003) 1–30. [Google Scholar]
  15. G. Colonna, On the relevance of superelastic collisions in argon and nitrogen discharges. Plasma Sources Sci. Technol. 29 (2020) 065008. [CrossRef] [Google Scholar]
  16. F. Diele and C. Marangi, Geometric numerical integration in ecological modelling. Mathematics 8 (2020) 25. [Google Scholar]
  17. B.A. Earnshaw and J.P. Keener, Global asymptotic stability of solutions of nonautonomous master equations. SIAM J. Appl. Dyn. Syst. 9 (2010) 220–237. [CrossRef] [MathSciNet] [Google Scholar]
  18. B.A. Earnshaw and J.P. Keener, Invariant manifolds of binomial-like nonautonomous master equations. SIAM J. Appl. Dyn. Syst. 9 (2010) 568–588. [CrossRef] [MathSciNet] [Google Scholar]
  19. L. Edsberg, Integration package for chemical kinetics. In: Stiff Differential Systems (Proc. Internat. Sympos., Wildbad, 1973), edited by R.A. Willoughby. Springer, Boston, MA (1974) 81–95. [Google Scholar]
  20. L. Formaggia and A. Scotti, Positivity and conservation properties of some integration schemes for mass action kinetics. SIAM J. Numer. Anal. 49 (2011) 1267–1288. [CrossRef] [MathSciNet] [Google Scholar]
  21. G. Giordano, F. Blanchini, R. Bruno, P. Colaneri, A. Di Filippo, A. Di Matteo and M. Colaneri, Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy. Nat. Med. 26 (2020) 855–860. [PubMed] [Google Scholar]
  22. J. Gunawardena, A linear framework for time-scale separation in nonlinear biochemical systems. PloS One 7 (2012) e36321. [CrossRef] [PubMed] [Google Scholar]
  23. O. Hadač, F. Muzika, V. Nevoral, M. Přibyl and I. Schreiber, Minimal oscillating subnetwork in the Huang-Ferrell model of the MAPK cascade. Plos One 12 (2017) e0178457. [CrossRef] [PubMed] [Google Scholar]
  24. E. Hairer and G. Wanner, Solving Ordinary Differential Equations. II. Stiff and Differential-Algebraic Problems, 2nd revised edition, paperback. Vol. 14 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin (2010). [Google Scholar]
  25. E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Reprint of the second (2006) edition. Vol. 31 of Springer Series in Computational Mathematics. Springer, Heidelberg (2010). [Google Scholar]
  26. E. Hansen, F. Kramer and A. Ostermann, A second-order positivity preserving scheme for semilinear parabolic problems. Appl. Numer. Math. 62 (2012) 1428–1435. [CrossRef] [MathSciNet] [Google Scholar]
  27. A. Hellander, J. Klosa, P. Lötstedt and S. MacNamara, Robustness analysis of spatiotemporal models in the presence of extrinsic fluctuations. SIAM J. Appl. Math. 77 (2017) 1157–1183. [CrossRef] [MathSciNet] [Google Scholar]
  28. M. Hochbruck, A. Ostermann and J. Schweitzer, Exponential Rosenbrock-type methods. SIAM J. Numer. Anal. 47 (2008/2009) 786–803. [CrossRef] [Google Scholar]
  29. A. Iserles, A First Course in the Numerical Analysis of Differential Equations, 2nd edition. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2009). [Google Scholar]
  30. A. Iserles and S. MacNamara, Applications of Magnus expansions and pseudospectra to Markov processes. Eur. J. Appl. Maths 30 (2019) 400–425. [CrossRef] [Google Scholar]
  31. A. Iserles and S.P. Nørsett, On the solution of linear differential equations in Lie groups. R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 357 (1999) 983–1019. [CrossRef] [MathSciNet] [Google Scholar]
  32. A. Iserles, H.Z. Munthe-Kaas, S.P. Nørsett and A. Zanna, Lie-Group Methods. In: Acta numerica, 2000. Vol. 9 of Acta Numer. Cambridge Univ. Press, Cambridge (2000) 215–365. [Google Scholar]
  33. W.O. Kermack and A.G. McKendrick, A contribution to the mathematical theory of epidemics. Proc. R. Soc. London 115 (1927) 700–721. [Google Scholar]
  34. S. Kopecz and A. Meister, On order conditions for modified Patankar–Runge–Kutta schemes. Appl. Numer. Math. 123 (2018) 159–179. [CrossRef] [MathSciNet] [Google Scholar]
  35. S. Kopecz and A. Meister, Unconditionally positive and conservative third order modified Patankar–Runge–Kutta discretizations of production-destruction systems. BIT Numer. Math. 58 (2018) 691–728. [CrossRef] [Google Scholar]
  36. S.C. Leite and R.J. Williams, A constrained Langevin approximation for chemical reaction networks. Ann. Appl. Prob. 29 (2019) 1541–1608. [CrossRef] [Google Scholar]
  37. S. MacNamara, Cauchy integrals for computational solutions of master equations. ANZIAM J. 56 (2015) 32–51. [CrossRef] [Google Scholar]
  38. S. MacNamara, A.M. Bersani, K. Burrage and R.B. Sidje, Stochastic chemical kinetics and the total quasi-steady-state assumption: application to the stochastic simulation algorithm and chemical master equation. J. Chem. Phys. 129 (2008) 095105. [CrossRef] [Google Scholar]
  39. S. MacNamara, K. Burrage and R.B. Sidje, Multiscale modeling of chemical kinetics via the master equation. Multiscale Modeling Simul. 6 (2008) 1146–1168. [CrossRef] [Google Scholar]
  40. S. MacNamara, B. Henry and W. Mclean, Fractional Euler limits and their applications. SIAM J. Appl. Math. 77 (2017) 447–469. [CrossRef] [MathSciNet] [Google Scholar]
  41. S. MacNamara, S. Blanes and A. Iserles, Simulation of bimolecular reactions: numerical challenges with the graph Laplacian. ANZIAM J. 61 (2020) C59–C74. [CrossRef] [Google Scholar]
  42. W. Magnus, On the exponential solution of differential equations for a linear operator. Comm. Pure Appl. Math. 7 (1954) 649–673. [CrossRef] [MathSciNet] [Google Scholar]
  43. P.K. Maini, T.E. Woolley, R.E. Baker, E.A. Gaffney and S.S. Lee, Turing’s model for biological pattern formation and the robustness problem. J. R. Soc. Interface Focus 2 (2012) 487–496. [CrossRef] [PubMed] [Google Scholar]
  44. A. Martiradonna, G. Colonna and F. Diele, GeCo: Geometric Conservative nonstandard schemes for biochemical systems. Appl. Numer. Math. 155 (2020s) 38–57. [CrossRef] [MathSciNet] [Google Scholar]
  45. I. Mirzaev and J. Gunawardena, Laplacian dynamics on general graphs. Bull. Math. Biol. 75 (2013) 2118–2149. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  46. P. Öffner and D. Torlo, Arbitrary high-order, conservative and positivity preserving Patankar-type deferred correction schemes. Appl. Numer. Math. 153 (2020) 15–34. [CrossRef] [MathSciNet] [Google Scholar]
  47. S.V. Patankar, Numerical Heat Transfer and Fluid Flow. Series in Computational Methods in Mechanics and Thermal Sciences. Hemisphere Pub. Corp., New York (1980). [Google Scholar]
  48. L. Qiao, R.B. Nachbar, I.G. Kevrekidis and S.Y. Shvartsman, Bistability and oscillations in the Huang-Ferrell model of MAPK signaling. PLoS Comput. Biol. 3 (2007) e184. [CrossRef] [Google Scholar]
  49. A. Sandu, Positive numerical integration methods for chemical kinetic systems. J. Comput. Phys. 170 (2001) 589–602. [CrossRef] [MathSciNet] [Google Scholar]
  50. J.M. Sanz-Serna and M.P. Calvo, Numerical Hamiltonian Problems. Vol. 7 of Applied Mathematics and Mathematical Computation. Chapman & Hall, London (1994). [CrossRef] [Google Scholar]
  51. M.J. Shon and A.E. Cohen, Mass action at the single-molecule level. J. Am. Chem. Soc. 134 (2012) 14618–14623. [CrossRef] [PubMed] [Google Scholar]
  52. R.L. Speth, W.H. Green, S. MacNamara and G. Strang, Balanced splitting and rebalanced splitting. SIAM J. Numer. Anal. 51 (2013) 3084–3105. [CrossRef] [MathSciNet] [Google Scholar]
  53. C. Timm, Random transition-rate matrices for the master equation. Phys. Rev. E 80 (2009) 021140. [CrossRef] [PubMed] [Google Scholar]

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