Uniform nonautonomous attractors under discretization. (Q1433931)

The authors study the behavior of attractors for nonautonomous ordinary differential equations under variable stepsize one-step numerical discretizations. This generalizes earlier results from the first author and \textit{J. Lorenz} [SIAM J. Numer. Anal. 23, 986--995 (1986; Zbl 0613.65083)] obtained in an autonomous framework. More precisely, nonautonomous ordinary differential equations in a skew-product setting with compact base space are considered, i.e., they are of the form \[ \dot{x}=f(\theta_tp,x)\leqno{(\ast)} \] with \(f:P\times{\mathbb R^d}\to{\mathbb R^d}\), \(P\) being a compact metric space and \((\theta_t)_{t\in{\mathbb R}}\) being a flow on \(P\). Beyond some Lipschitz condition on \(f\), it is assumed that \((\ast)\) has a uniform pullback attractor \(\widehat{A}\). As main result, the authors show that the discrete skew-product system generated by a numerical variable stepsize discretization of \((\ast)\) also possesses a uniform pullback attractor \(\widehat{A}^\delta\) converging to \(\widehat{A}\) in the Hausdorff semimetric for \(\delta\to 0+\), provided the maximal stepsize \(\delta>0\) is sufficiently small. The main tools in the proof are a technique to write numerical schemes as discrete skew-product flows and a converse theorem for Lyapunov functions characterizing the uniform pullback attractor.