The Global Attractor Under Discretisation

Abstract

The effect of temporal discretisation on dissipative differential equations is analysed. We discuss the effect of discretisation on the global attractor and survey some recent results in the area. The advantage of concentrating on ω and α limit sets (which are contained in the global attractor) is described. An analysis of spurious bifurcations in the ω and α limit sets is presented for linear multistep methods, using the time-step Δtas the bifurcation parameter. The results arising from application of local bifurcation theory are shown to hold globally and a necessary and sufficient condition is derived for the non-existence of a particular class of spurious solutions, for allΔt> 0. The class of linear multistep methods satisfying this condition is fairly restricted since the underlying theory is very general and takes no account of any inherent structure in the underlying differential equations. Hence a method complementary to the bifurcation analysis is described, the aim being to construct methods for which spurious solutions do not exist forΔt sufficiently small; for infinite dimensional dynamical systems the method relies on examining steady boundary value problems (which govern the existence of spurious solutions) in the singular limit correpsonding to Δt→ 0₊. The analysis we describe is helpful in the design of schemes for long-time simulations.