On the linear stability of some finite difference schemes for nonlinear reaction-diffusion models of chemical reaction networks

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We identify sufficient conditions for the stability of some well-known finite difference schemes for the solution of the multivariable reaction-diffusion equations that model chemical reaction networks. Since the equations are mainly nonlinear, these conditions are obtained through local linearization. A recurrent condition is that the Jacobian matrix of the reaction part evaluated at some positive unknown solution is either D-semi-stable or semi-stable. We demonstrate that for a single reversible chemical reaction whose kinetics are monotone, the Jacobian matrix is D-semi-stable and therefore such schemes are guaranteed to work well.

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  • 1. R. Fisher The wave of advance of advantageous genes Ann Hum Genet vol. 7 pp. 355{369 1937.

  • 2. A. D. Bazykin Hypothetical mechanism of speciation Evolution vol. 23 pp. 685{687 1969.

  • 3. B. Bradshaw-Hajek Reaction-di usion equations for population genetics. PhD thesis School of Math- ematics and Applied Statistics University of Wollongong 2004.

  • 4. J. G. Skellam Random dispersal in theoretical populations Biometrika vol. 38 pp. 196{218 1951.

  • 5. A. M. Turing The chemical basis of morphogenesis Phil. Trans. R. Soc. Lond. pp. 37{72 1952.

  • 6. A. Gierer and H. Meinhardt A theory of biological pattern formation kybernetik vol. 12 pp. 30{39 1972.

  • 7. J. D. Murray Mathematical Biology: I. An Introduction. Springer 1989.

  • 8. J. D. Murray Mathematical Biology: II. Spatial Models and Biomedical Applications. Springer 2003.

  • 9. J. J. Tyson and P. C. Fife Target patterns in a realistic model of the belousov-zhabotinskii reaction J. Chem. Phys. vol. 73 pp. 2224{2237 1980.

  • 10. J. Sneyd A. C. Charles and M. J. Sanderson A model for the propagation of intracellular calcium waves Am. J. Physiol. vol. 266 pp. 293{302 1994.

  • 11. J. Sneyd B. T. R. Wetton A. C. Charles and M. J. Sanderson Intracellular calcium waves mediated by diffusion of inositol triphosphate; a two dimensional model Am. J. Physiol. vol. 268 pp. 1537-1545 1995.

  • 12. S. Means A. J. Smith J. Shepherd J. Shadid J. Fowler R. J. H. Wojcikiewicz T. Mazel G. D. Smith and B. S. Wilson Reaction-diffusion modelling of calcium dynamics with realistic er geometry Biophysical Journal vol. 91 pp. 537{557 2006.

  • 13. X.-S. Yang Computational modelling of nonlinear calcium waves Appl. Math. Model. vol. 30 pp. 200{208 2006.

  • 14. M. A. Colman C. Pinali A. W. Trafford H. Zhang and A. Kitmitto A computational model of spatio- temporal cardiac intracellular calcium handling with realistic structure and spatial flux distribution from sarcoplasmic reticulum and t-tubule reconstructions PLOS Comput. Biol. vol. 13 2017.

  • 15. E. Meron Pattern formation in excitable media Physics Reports vol. 218 no. 1 pp. 1 { 66 1992.

  • 16. J. M. Hyman The method of lines solution of partial differential equations Tech. Rep. C00-3077-139 New York University October 1976.

  • 17. L. Brugnano F. Mazzia and D. Trigiante Fifty years of stiffness in Recent Advances in Computational and Applied Mathematics (T. E. Simos ed.) ch. 1 pp. 1{21 Springer Netherlands 2011.

  • 18. K. Radhakrishnan Comparison of numerical techniques for integration of stiff ordinary differential equations arising in combustion chemistry Tech. Rep. NASA-TP-2372 NASA October 1984.

  • 19. J. Crank and P. Nicolson A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type Adv. Comput. Math. vol. 6 pp. 207{226 1996.

  • 20. R. Glowinski Finite element methods for incompressible viscous ow Handbook of Numerical Anal- ysis vol. 9 pp. 3{1176 2003.

  • 21. A. Madzvamuse and A. H. Chung Fully implicit time-stepping schemes and non-linear solvers for systems of reaction diffusion equations Appl. Math. Comput. vol. 244 pp. 361{374 2014.

  • 22. A.-K. Kassam and L. N. Trefethen Fourth-order time-stepping for stiff pdes SIAM J. Sci. Comput. vol. 26 no. 4 pp. 1214{1233 2005.

  • 23. U. M. Ascher S. J. Ruuth and B. T. R. Wetton Implicit-explicit methods for time-dependent partial differential equations SIAM J. Numer. Anal. vol. 32 no. 3 pp. 797{823 1995.

  • 24. H.Wang C.-W. Shu and Q. Zhang Stability and error estimates of local discontinuous galerkin meth- ods with implicit-explicit time-marching for advection-diffusion problems SIAM J. Numer. Anal. vol. 53 pp. 206{227 2015.

  • 25. A. Madzvamuse Time-stepping schemes for moving grid finite elements applied to reaction-diffusion systems on fixed and growing domains J. Comput. Phys. vol. 214 pp. 239-263 2006.

  • 26. Q. Nie Y.-T. Wan Frederic Y. M.and Zhang and X.-F. Liu Compact integration factor methods in high spatial dimensions J. Comput. Phys. vol. 227 pp. 5238{5255 2008.

  • 27. Q. Nie Y.-T. Zhang and Z. Rui Efficient semi-implicit schemes for stiff systems J. Comput. Phys. vol. 214 pp. 521{537 2006.

  • 28. J. W. Thomas Numerical Partial Differential Equations: Finite Difference Methods. Springer 1995.

  • 29. D. Ho Stability and convergence of finite difference methods for systems of nonlinear reaction- diffusion equations SIAM J. Numer. Anal. vol. 15 no. 6 pp. 1161{1177 1978.

  • 30. A. Araújo S. Barbeiro and P. Serranho Stability of finite difference schemes for complex diffusion processes SIAM J. Numer. Anal. vol. 50 no. 3 pp. 1284-1296 2012.

  • 31. A. Araújo S. Barbeiro and P. Serranho Stability of finite difference schemes for nonlinear complex reaction-diffusion processes IMA J. Numer. Anal. vol. 35 pp. 1381-1401 2015.

  • 32. N. Li J. Steiner and S. Tang Convergence and stability analysis of an explicit finite difference method for 2-dimensional reaction-diffusion equations J. Austral. Math. Soc. vol. 36 pp. 234{241 1994.

  • 33. C. Hirsch Numerical Computation of Internal and External Flows: Fundamentals of Numerical Discretization. John Wiley & Sons 1988.

  • 34. R. A. Horn and C. R. Johnson Matrix Analysis. Cambridge 2013.

  • 35. R. D. Richtmyer and K. W. Morton Difference Methods for Initial-Value Problems. John Wiley & Sons 1967.

  • 36. D. Serre Matrices; Theory and Applications. Springer 2010.

  • 37. A. J. Laub Matrix Analysis for Scientists and Engineers. SIAM 2005.

  • 38. B. Dasgupta Applied Mathematical Methods. Dorling Kindersley (India) Pvt. Ltd. 2006.

  • 39. D. Siegel Chemical reaction networks as compartmental systems Presented at the 21st International Symposium on Mathematical Theory of Networks and Systems July 7-11 2014. Groningen The Netherlands.

  • 40. G. Giorgi and C. Zuccotti An overview on d-stable matrices. Universita Di Pavia DEM Working Paper Series Feb. 2015.

  • 41. G. Nicolis and I. Prigogine Self-organization in nonequilibrium systems : from dissipative structures to order through fluctuations / g. nicolis i. prigogine 01 1977.

  • 42. I. Prigogine and R. Lefever Symmetry breaking instabilities in dissipative systems. ii The Journal of Chemical Physics vol. 48 no. 4 pp. 1695{1700 1968.

  • 43. M. Marek and I. Schreiber Chaotic behaviour of deterministic dissipative systems. Cambridge University Press 1991.

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