Inertial manifolds of parabolic differential equations under high-order discretizations

This paper deals with the long-time behaviour of numerical discretizations of nonlinear parabolic differential equations. For various equations of mathematical physics, the dynamics are governed by a finite-dimensional inertial manifold, which attracts solutions at an exponential rate. We show that Runge-Kutta time and spectral Galerkin space discretizations possess inertial manifolds which approximate the inertial manifold of the continuous problem with the order of finite-time approximations of smooth solutions. We thus 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 convergence of the space discretization. These results are obtained by using time analyticity and Gevrey regularity of solutions of the differential equation. As an application of the theory, the complex Ginzburg-Landau equation is considered.

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