Abstract
In this work, we introduce a simple and efficient approach for constructing dynamically consistent and high-order nonstandard finite difference (NSFD) methods for a class of nonlinear dynamical systems. These high-order NSFD methods are derived from a novel weighted non-local approximation of the right-hand side functions and the renormalization of nonstandard denominator functions. It is proved by rigorous mathematical analysis that the NSFD methods can be accurate of arbitrary order and preserve two essential mathematical features including the positivity and asymptotic stability of the dynamical systems under consideration for all finite values of the step size. In the constructed NSFD methods, the weighted non-local approximation guarantees the dynamic consistency and the renormalized nonstandard denominator functions ensure that the proposed NSFD methods are accurate of arbitrary order. The obtained results resolve the contradiction between the dynamic consistency and high-order accuracy of NSFD schemes and hence, improve several well-known results focusing on NSFD methods for differential equations. Finally, illustrative numerical examples are reported to support the theoretical findings and to show advantages of the high-order NSFD methods.
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References
Allen, L.J.S.: An Introduction to Mathematical Biology. Prentice Hall (2007)
Anguelov, R., Lubuma, J.M.-S.: Contributions to the mathematics of the nonstandard finite difference method and applications. Numer. Methods Partial Differ. Equ. 17, 518–543 (2001)
Anguelov, R., Lubuma, J.M.-S.: Nonstandard finite difference method by nonlocal approximation. Math. Comput. Simul. 61, 465–475 (2003)
Ascher, U.M., Petzold, L.R.: Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. Society for Industrial and Applied Mathematics, Philadelphia (1998)
Burden, R.L., Douglas Faires, F.: Numerical Analysis, Ninth edition Cengage Learning (2015)
Chen-Charpentier, B.M., Dimitrov, D.T., Kojouharov, H.V.: Combined nonstandard numerical methods for ODEs with polynomial right-hand sides. Math. Comput. Simul. 73, 105–113 (2006)
Cooke, K., van den Driessche, P., Zou, X.: Interaction of maturation delay and nonlinear birth in population and epidemic models. J. Math. Biol. 39, 332–352 (1999)
Cresson, J., Pierret, F.: Non standard finite difference scheme preserving dynamical properties. J. Comput. Appl. Math. 303, 15–30 (2016)
Cresson, J., Szafrańska, A.: Discrete and continuous fractional persistence problems - the positivity property and applications. Commun. Nonlinear Sci. Numer. Simul. 44, 424–448 (2017)
Dang, Q.A., Hoang, M.T.: Positive and elementary stable explicit nonstandard Runge-Kutta methods for a class of autonomous dynamical systems. Int. J. Comput. Math. 97, 2036–2054 (2020)
Dimitrov, D.T., Kojouharov, H.V.: Nonstandard finite-difference schemes for general two-dimensional autonomous dynamical systems. Appl. Math. Lett. 18, 769–774 (2005)
Elaydi, S.: An Introduction to Difference Equations. Springer, New York (2005)
Fatoorehchi, H., Ehrhardt, M.: Numerical and semi-numerical solutions of a modified Thévenin model for calculating terminal voltage of battery cells. J. Energy Storag 45, 103746 (2022)
Gonzalez-Parra, G., Arenas, A.J., Chen-Charpentier, B.M.: Combination of nonstandard schemes and Richardson’s extrapolation to improve the numerical solution of population models. Math. Comput. Model. 52, 1030–1036 (2010)
Gupta, M., Slezak, J.M., Alalhareth, F., Roy, S., Kojouharov, H.V.: Second-order nonstandard explicit Euler method. AIP Conf. Proc 2302, 110003 (2020)
Hoang, M.T.: A novel second-order nonstandard finite difference method for solving one-dimensional autonomous dynamical systems. Commun. Nonlinear Sci. Numer. Simul. 114, 106654 (2022)
Hoang, M.T.: A class of second-order and dynamically consistent nonstandard finite difference schemes for nonlinear Volterra’s population growth model. Comput. Appl. Math. 42, 85 (2023)
Hoang, M.T.: A novel second-order nonstandard finite difference method preserving dynamical properties of a general single-species model. Int. J. Comput. Math. 100, 2047–2062 (2023)
Hoang, M.T., Ehrhardt, M.: A second-order nonstandard finite difference method for a general Rosenzweig-MacArthur predator-prey model. J. Comput. Appl. Math. 44, 115752 (2024)
Hoang, M.T., Ehrhardt, M.: A general class of second-order \(L\)-stable explicit numerical methods for stiff problems. Appl. Math. Lett. 149, 108897 (2024)
Horváth, Z.: Positivity of Runge-Kutta and diagonally split Runge-Kutta methods, AApplied. Numer. Math. 28, 309–326 (1998)
Jiang, Z., Zhang, W.: Bifurcation analysis in single-species population model with delay. Sci. China Math. 53, 1475–1481 (2010)
Kojouharov, H.V., Roy, S., Gupta, M., Alalhareth, F., Slezak, J.M.: A second-order modified nonstandard theta method for one-dimensional autonomous differential equations. Appl. Math. Lett. 112, 106775 (2021)
Martin-Vaquero, J., Martin del Rey, A., Encinas, A.H., Hernandez Guillen, J.D., Queiruga-Dios, A., Rodriguez Sanchez, G.: Higher-order nonstandard finite difference schemes for a MSEIR model for a malware propagation. J. Comput. Appl. Math. 317, 146–156 (2017)
Martin-Vaquero, J., Queiruga-Dios, A., Martin del Rey, A., Encinas, A.H., Hernandez Guillen, J.D., Rodriguez Sanchez, G.: Variable step length algorithms with high-order extrapolated non-standard finite difference schemes for a SEIR model. J. Comput. Appl. Math. 330, 848–854 (2018)
Mickens, R.E.: Nonstandard Finite Difference Models of Differential Equations, World Scientific,(1993)
Mickens, R. E.: Applications of Nonstandard Finite Difference Schemes, World Scientific, (2000)
Mickens, R. E.: Advances in the Applications of Nonstandard Finite Difference Schemes, World Scientific, (2005)
Mickens, R. E.: Nonstandard Finite Difference Schemes: Methodology and Applications, World Scientific, (2020)
Mickens, R.E.: Dynamic consistency: a fundamental principle for constructing nonstandard finite difference schemes for differential equations. J. Differ. Equ. Appl. 11, 645–653 (2005)
Mickens, R.E.: Nonstandard finite difference schemes for differential equations. J. Differ. Equ. Appl. 8, 823–847 (2002)
Mickens, R.E.: Discretizations of nonlinear differential equations using explicit nonstandard methods. J. Comput. Appl. Math. 110, 181–185 (1999)
Mickens, R.E., Washington, T.M.: NSFD discretizations of interacting population models satisfying conservation laws. Comput. Math. Appl. 66, 2307–231 (2013)
Patidar, K.C.: On the use of nonstandard finite difference methods. J. Differ. Equ. Appl. 11, 735–758 (2005)
Patidar, K.C.: Nonstandard finite difference methods: recent trends and further developments. J. Differ. Equ. Appl. 22, 817–849 (2016)
Smith, H. L., Waltman, P.: The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, (1995)
Stuart, A., Humphries, A. R.: Dynamical systems and numerical analysis, Cambridge University Press, (1998)
Sun, Z., Lv, J., Zou, X.: Dynamical analysis on two stochastic single-species models. Appl. Math. Lett. 99, 105982 (2020)
Wood, D.T., Kojouharov, H.V.: A class of nonstandard numerical methods for autonomous dynamical systems. Appl. Math. Lett. 50, 78–82 (2015)
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I would like to thank the editor and the anonymous reviewer and referee for their useful and valuable comments, which led to a great improvement of the paper.
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Manh Tuan Hoang: conceptualization, methodology, software, formal analysis, writing — original draft preparation, methodology, writing — review and editing, supervision.
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Hoang, M.T. High-order nonstandard finite difference methods preserving dynamical properties of one-dimensional dynamical systems. Numer Algor (2024). https://doi.org/10.1007/s11075-024-01792-1
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DOI: https://doi.org/10.1007/s11075-024-01792-1