Discretization with Dense Uncountable Continuity for Global Attractors of Kuramoto-Sivashinsky Lattice Equations

We study both continuous- and discrete-time Kuramoto-Sivashinsky lattice equations with an anti-dissipative linearity and a polynomial damping. We estimate the uniform bound of Taylor remainder terms for continuous-time solutions, which allows us to establish the interpolation error and the convergence rate between continuous- and discrete-time solutions. We then prove the existence of a family of numerical attractors and establish the upper semicontinuous convergence of the family towards the original global attractor as the stepsize approaches zero. We also show the upper semicontinuous stability of the numerical attractors perturbed by the finitely dimension attractors. More importantly, we prove that the set of all continuous points of numerical attractors under Hausdorff’s distance is residual, dense and uncountable in the existing interval of stepsizes.