Heriot-Watt Mathematics Report Series
HWM01-24, 12 Jul 2001

Topological and $\epsilon$-entropy for Large Volume Limits of Discretised Parabolic Equations

G J Lord and J Rougemont

Abstract

We consider semi-discrete and fully discrete approximations of nonlinear parabolic equations in the limit of unbounded domains, which by a scaling argument is equivalent to the limit of small viscosity. We define the spatial density of $\epsilon $-entropy, topological entropy and dimension for the attractors and show that these quantities are bounded. We also provide practical means of computing lower bounds on them. The proof uses the property that solutions lie in Gevrey classes of analyticity, which we define in a way that does not depend on the size of the spatial domain. As a specific example we discuss the complex Ginzburg-Landau equation.

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Publication

G J Lord, A N Smith , Topological and $\epsilon$-entropy for large volume limits of discretised parabolic equations, Siam J Num Anal, 40, 1311-1329, (2002).

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