Inertial manifolds of parabolic differential equations under higher-order discretizations

This paper deals with the long-time behaviour of numerical discretizations of nonlinear parabolic differential equations. For various equations of mat hematical physics, the dynamics are governed by a finite-dimensional inerti al manifold, which attracts solutions at an exponential rate. We show that Runge-Kutta time and spectral Galerkin space discretizations possess inerti al manifolds which approximate the inertial manifold of the continuous prob lem with the order of finite-time approximations of smooth solutions. We th us obtain estimates for the distance between the inertial manifolds of the partial differential equation and its semi- and full discretizations which show the high order of the time discretization and exponentially fast conve rgence of the space discretization. These results are obtained by using tim e analyticity and Gevrey regularity of solutions of the differential equati on. As an application of the theory, the complex Ginzburg-Landau equation i s considered.