Design and analysis of NSFD methods for the diffusion-free Brusselator. (English) Zbl 1330.65118
Gumel, Abba B. (ed.), Mathematics of continuous and discrete dynamical systems. AMS special session in honor of Ronald Mickens’s 70th birthday on nonstandard finite-difference discretizations and nonlinear oscillations, San Diego, CA, USA, January 9–10, 2013. Proceedings. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-9862-8/pbk; 978-1-4704-1686-7/ebook). Contemporary Mathematics 618, 163-180 (2014).
Summary: This chapter reports on the design and analysis of two non-standard finite-difference (NSFD) methods for solving the diffusion-free Brusselator system. The first NSFD method is simulated using two different denominator functions. It is shown that, under certain conditions, the first method exhibits spurious behaviour, such as failing to capture the correct asymptotic stability properties of the unique fixed-point as well as the stable limit cycle of the continuous-time diffusion-free Brusselator system. On the other hand, the second NSFD method, designed using a semi-exact discretization framewoek of R. E. Mickens [Nonstandard finite difference models of differential equations. Singapore: World Scientific (1994; Zbl 0810.65083)] is shown to be elementary stable and dynamically consistent with the diffusion-free Brusselator system for sufficiently small timesteps. These theoretical results are illustrated via numerical simulation.
For the entire collection see [Zbl 1294.34001].
For the entire collection see [Zbl 1294.34001].
MSC:
| 65L12 | Finite difference and finite volume methods for ordinary differential equations |
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 34A34 | Nonlinear ordinary differential equations and systems |
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
