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The Global Dynamics of Discrete Semilinear Parabolic Equations

Elliott, C. M. and Stuart, A. M. (1993) The Global Dynamics of Discrete Semilinear Parabolic Equations. SIAM Journal on Numerical Analysis, 30 (6). pp. 1622-1663. ISSN 0036-1429.

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A class of scalar semilinear parabolic equations possessing absorbing sets, a Lyapunov functional, and a global attractor are considered. The gradient structure of the problem implies that, provided all steady states are isolated, solutions approach a steady state as $t \to \infty $. The dynamical properties of various finite difference and finite element schemes for the equations are analysed. The existence of absorbing sets, bounded independently of the mesh size, is proved for the numerical methods. Discrete Lyapunov functions are constructed to show that, under appropriate conditions on the mesh parameters, numerical orbits approach steady state solutions as discrete time increases. However, it is shown that insufficient spatial resolution can introduce deceptively smooth spurious steady solutions and cause the stability properties of the true steady solutions to be incorrectly represented. Furthermore, it is also shown that the explicit Euler scheme introduces spurious solutions with period 2 in the timestep. As a result, the absorbing set is destroyed and there is initial data leading to blow up of the scheme, however small the mesh parameters are taken. To obtain stabilization to a steady state for this scheme, it is necessary to restrict the timestep in terms of the initial data and the space step. Implicit schemes are constructed for which absorbing sets and Lyapunov functions exist under restrictions on the timestep that are independent of initial data and of the space step; both one-step and multistep (BDF) methods are studied.

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Additional Information:© 1993 Society for Industrial and Applied Mathematics. Submitted: 14 April 1992. Accepted: 28 December 1992. A preliminary version of this work was presented at the IMA (UK) "Dynamics of Numerics and Numerics of Dynamics" conference, July 1990, Bristol. The work of C. M. Elliott was partially supported by the Institute of Mathematics and its Applications (Minneapolis) with funds provided by the National Science Foundation during the 1990/91 program on "Phase Transitions and Free Boundaries." The authors are grateful to Arieh Iserles for helpful discussions regarding the material in 3.1.
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Subject Keywords:attractors, absorbing sets, Lyapunov functions, spurious solutions
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Andrew StuartJ22
Issue or Number:6
Classification Code:AMS subject classifications. 65M99, 35K57
Record Number:CaltechAUTHORS:20170613-070150162
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Official Citation:The Global Dynamics of Discrete Semilinear Parabolic Equations C. M. Elliott and A. M. Stuart SIAM Journal on Numerical Analysis 1993 30:6, 1622-1663
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:78136
Deposited By: Ruth Sustaita
Deposited On:13 Jun 2017 16:17
Last Modified:03 Oct 2019 18:05

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