American Institute of Mathematical Sciences

Advanced
2009, 2009(Special): 34-43. doi: 10.3934/proc.2009.2009.34

Dynamically consistent nonstandard finite difference schemes for continuous dynamical systems

Roumen Anguelov 1, , Jean M.-S. Lubuma 1, and  Meir Shillor 2,

1. 

Department of Mathematics and Applied Mathematics, University of Pretoria, South Africa, South Africa
2. 

Department of Mathematics and Statistics, Oakland University, Rochester, MI 48309

Received  July 2008 Revised  March 2009 Published  September 2009

This work deals with the relationship between a continuous dynamical system and numerical methods for its computer simulations, viewed as discrete dynamical systems. The term 'dynamic consistency' of a numerical scheme with the associated continuous system is usually loosely defined, meaning that the numerical solutions replicate some of the properties of the solutions of the continuous system. Here, this concept is replaced with topological dynamic consistency, which is defined in precise terms through the topological equivalence of maps. This ensures that all the topological properties (e.g., fixed points and their stability, periodic solutions, invariant sets, etc.) are preserved. Two examples are provided which demonstrate that numerical schemes satisfying this strong notion of dynamic consistency can be constructed using the nonstandard finite difference method.
Keywords: nonstandard finite difference method, dynamical systems, dynamic consistency.
Mathematics Subject Classification: Primary: 65L12; Secondary: 37B99.
Citation: Roumen Anguelov, Jean M.-S. Lubuma, Meir Shillor. Dynamically consistent nonstandard finite difference schemes for continuous dynamical systems. Conference Publications, 2009, 2009 (Special) : 34-43. doi: 10.3934/proc.2009.2009.34
[1]

Ronald E. Mickens. A nonstandard finite difference scheme for the drift-diffusion system. Conference Publications, 2009, 2009 (Special) : 558-563. doi: 10.3934/proc.2009.2009.558
[2]

Lih-Ing W. Roeger. Dynamically consistent discrete Lotka-Volterra competition models derived from nonstandard finite-difference schemes. Discrete and Continuous Dynamical Systems - B, 2008, 9 (2) : 415-429. doi: 10.3934/dcdsb.2008.9.415
[3]

Junxiang Li, Yan Gao, Tao Dai, Chunming Ye, Qiang Su, Jiazhen Huo. Substitution secant/finite difference method to large sparse minimax problems. Journal of Industrial and Management Optimization, 2014, 10 (2) : 637-663. doi: 10.3934/jimo.2014.10.637
[4]

Hongsong Feng, Shan Zhao. A multigrid based finite difference method for solving parabolic interface problem. Electronic Research Archive, 2021, 29 (5) : 3141-3170. doi: 10.3934/era.2021031
[5]

Brittany Froese Hamfeldt, Jacob Lesniewski. A convergent finite difference method for computing minimal Lagrangian graphs. Communications on Pure and Applied Analysis, 2022, 21 (2) : 393-418. doi: 10.3934/cpaa.2021182
[6]

Igor E. Shparlinski. On some dynamical systems in finite fields and residue rings. Discrete and Continuous Dynamical Systems, 2007, 17 (4) : 901-917. doi: 10.3934/dcds.2007.17.901
[7]

Liejune Shiau, Roland Glowinski. Operator splitting method for friction constrained dynamical systems. Conference Publications, 2005, 2005 (Special) : 806-815. doi: 10.3934/proc.2005.2005.806
[8]

Moulay Rchid Sidi Ammi, Ismail Jamiai. Finite difference and Legendre spectral method for a time-fractional diffusion-convection equation for image restoration. Discrete and Continuous Dynamical Systems - S, 2018, 11 (1) : 103-117. doi: 10.3934/dcdss.2018007
[9]

Weizhu Bao, Chunmei Su. Uniform error estimates of a finite difference method for the Klein-Gordon-Schrödinger system in the nonrelativistic and massless limit regimes. Kinetic and Related Models, 2018, 11 (4) : 1037-1062. doi: 10.3934/krm.2018040
[10]

Ömer Oruç, Alaattin Esen, Fatih Bulut. A unified finite difference Chebyshev wavelet method for numerically solving time fractional Burgers' equation. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 533-542. doi: 10.3934/dcdss.2019035
[11]

Wen Chen, Song Wang. A finite difference method for pricing European and American options under a geometric Lévy process. Journal of Industrial and Management Optimization, 2015, 11 (1) : 241-264. doi: 10.3934/jimo.2015.11.241
[12]

Yones Esmaeelzade Aghdam, Hamid Safdari, Yaqub Azari, Hossein Jafari, Dumitru Baleanu. Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact finite difference scheme. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2025-2039. doi: 10.3934/dcdss.2020402
[13]

Takeshi Fukao, Shuji Yoshikawa, Saori Wada. Structure-preserving finite difference schemes for the Cahn-Hilliard equation with dynamic boundary conditions in the one-dimensional case. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1915-1938. doi: 10.3934/cpaa.2017093
[14]

Jianjun Paul Tian. Finite-time perturbations of dynamical systems and applications to tumor therapy. Discrete and Continuous Dynamical Systems - B, 2009, 12 (2) : 469-479. doi: 10.3934/dcdsb.2009.12.469
[15]

Xavier Cabré, Amadeu Delshams, Marian Gidea, Chongchun Zeng. Preface of Llavefest: A broad perspective on finite and infinite dimensional dynamical systems. Discrete and Continuous Dynamical Systems, 2018, 38 (12) : i-iii. doi: 10.3934/dcds.201812i
[16]

María J. Garrido-Atienza, Oleksiy V. Kapustyan, José Valero. Preface to the special issue "Finite and infinite dimensional multivalued dynamical systems". Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : i-iv. doi: 10.3934/dcdsb.201705i
[17]

Tianhu Yu, Jinde Cao, Chuangxia Huang. Finite-time cluster synchronization of coupled dynamical systems with impulsive effects. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3595-3620. doi: 10.3934/dcdsb.2020248
[18]

Felix X.-F. Ye, Hong Qian. Stochastic dynamics Ⅱ: Finite random dynamical systems, linear representation, and entropy production. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4341-4366. doi: 10.3934/dcdsb.2019122
[19]

Tomasz R. Bielecki, Igor Cialenco, Marcin Pitera. A survey of time consistency of dynamic risk measures and dynamic performance measures in discrete time: LM-measure perspective. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 3-. doi: 10.1186/s41546-017-0012-9
[20]

Hawraa Alsayed, Hussein Fakih, Alain Miranville, Ali Wehbe. Finite difference scheme for 2D parabolic problem modelling electrostatic Micro-Electromechanical Systems. Electronic Research Announcements, 2019, 26: 54-71. doi: 10.3934/era.2019.26.005

 Impact Factor: 

Tools

Metrics

  • PDF downloads (192)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy