A holistic finite difference approach models linear dynamics consistently

Author:
A. J. Roberts

Journal:
Math. Comp. **72** (2003), 247-262

MSC (2000):
Primary 37L65, 65M20, 37L10, 65P40, 37M99

Published electronically:
June 4, 2002

MathSciNet review:
1933820

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Abstract | References | Similar Articles | Additional Information

Abstract: I prove that a centre manifold approach to creating finite difference models will consistently model linear dynamics as the grid spacing becomes small. Using such tools of dynamical systems theory gives new assurances about the quality of finite difference models under nonlinear and other perturbations on grids with finite spacing. For example, the linear advection-diffusion equation is found to be stably modelled for all advection speeds and all grid spacings. The theorems establish an extremely good form for the artificial internal boundary conditions that need to be introduced to apply centre manifold theory. When numerically solving nonlinear partial differential equations, this approach can be used to systematically derive finite difference models which automatically have excellent characteristics. Their good performance for finite grid spacing implies that fewer grid points may be used and consequently there will be less difficulties with stiff rapidly decaying modes in continuum problems.

**1.**Jack Carr,*Applications of centre manifold theory*, Applied Mathematical Sciences, vol. 35, Springer-Verlag, New York-Berlin, 1981. MR**635782****2.**Jack Carr and Robert G. Muncaster,*The application of centre manifolds to amplitude expansions. II. Infinite-dimensional problems*, J. Differential Equations**50**(1983), no. 2, 280–288. MR**719450**, 10.1016/0022-0396(83)90078-5**3.**P. Constantin, C. Foias, B. Nicolaenko, and R. Temam,*Integral manifolds and inertial manifolds for dissipative partial differential equations*, Applied Mathematical Sciences, vol. 70, Springer-Verlag, New York, 1989. MR**966192****4.**Stephen M. Cox and A. J. Roberts,*Initial conditions for models of dynamical systems*, Phys. D**85**(1995), no. 1-2, 126–141. MR**1339235**, 10.1016/0167-2789(94)00201-Z**5.**C. Foias, M. S. Jolly, I. G. Kevrekedis, G. R. Sell, and E. S. Titi.

On the computation of inertial manifolds.*Phys Lett. A*, 131:433-436, 1988.**6.**C. Foias, M. S. Jolly, I. G. Kevrekidis, G. R. Sell, and E. S. Titi,*On the computation of inertial manifolds*, Phys. Lett. A**131**(1988), no. 7-8, 433–436. MR**972615**, 10.1016/0375-9601(88)90295-2**7.**Ciprian Foias, George R. Sell, and Edriss S. Titi,*Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations*, J. Dynam. Differential Equations**1**(1989), no. 2, 199–244. MR**1010966**, 10.1007/BF01047831**8.**Ciprian Foias and Edriss S. Titi,*Determining nodes, finite difference schemes and inertial manifolds*, Nonlinearity**4**(1991), no. 1, 135–153. MR**1092888****9.**Bosco García-Archilla and Edriss S. Titi,*Postprocessing the Galerkin method: the finite-element case*, SIAM J. Numer. Anal.**37**(2000), no. 2, 470–499. MR**1740770**, 10.1137/S0036142998335893**10.**M. S. Jolly, I. G. Kevrekidis, and E. S. Titi,*Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: analysis and computations*, Phys. D**44**(1990), no. 1-2, 38–60. MR**1069671**, 10.1016/0167-2789(90)90046-R**11.**T. Mackenzie and A. J. Roberts.

Holistic finite differences accurately model the dynamics of the Kuramoto-Sivashinsky equation.*ANZIAM J.*, 42(E):C918-C935, 2000.

[Online]`http://anziamj.austms.org.au/V42/CTAC99/Mack`.**12.**Martine Marion and Roger Temam,*Nonlinear Galerkin methods*, SIAM J. Numer. Anal.**26**(1989), no. 5, 1139–1157. MR**1014878**, 10.1137/0726063**13.***Modern computing methods*, 2nd ed. National Physical Laboratory. Notes on Applied Science, No. 16, Her Majesty’s Stationery Office, London, 1961. MR**0117863****14.**A. J. Roberts,*Appropriate initial conditions for asymptotic descriptions of the long term evolution of dynamical systems*, J. Austral. Math. Soc. Ser. B**31**(1989), no. 1, 48–75. MR**1002091**, 10.1017/S0334270000006470**15.**A. J. Roberts,*The utility of an invariant manifold description of the evolution of a dynamical system*, SIAM J. Math. Anal.**20**(1989), no. 6, 1447–1458. MR**1019310**, 10.1137/0520094**16.**A. J. Roberts,*Low-dimensional models of thin film fluid dynamics*, Phys. Lett. A**212**(1996), no. 1-2, 63–71. MR**1379007**, 10.1016/0375-9601(96)00040-0**17.**A. J. Roberts.

Computer algebra derives correct initial conditions for low-dimensional dynamical models.*Comput. Phys. Comm.*, 126(3):187-206, 2000.**18.**A. J. Roberts.

Holistic discretisation ensures fidelity to Burgers' equation.*Applied Numerical Math.*, 37:371-396, 2001.**19.**James C. Robinson,*The asymptotic completeness of inertial manifolds*, Nonlinearity**9**(1996), no. 5, 1325–1340. MR**1416479**, 10.1088/0951-7715/9/5/013**20.**R. Temam,*Inertial manifolds*, Math. Intelligencer**12**(1990), no. 4, 68–74. MR**1076537**, 10.1007/BF03024036**21.**A. Vanderbauwhede,*Centre manifolds, normal forms and elementary bifurcations*, Dynamics reported, Vol. 2, Dynam. Report. Ser. Dynam. Systems Appl., vol. 2, Wiley, Chichester, 1989, pp. 89–169. MR**1000977**

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Additional Information

**A. J. Roberts**

Affiliation:
Department of Mathematics and Computing, University of Southern Queensland, Toowoomba, Queensland 4352, Australia

Email:
aroberts@usq.edu.au

DOI:
https://doi.org/10.1090/S0025-5718-02-01448-5

Received by editor(s):
April 6, 2000

Received by editor(s) in revised form:
November 14, 2000

Published electronically:
June 4, 2002

Article copyright:
© Copyright 2002
American Mathematical Society