A finite difference scheme for computing inertial manifolds

A fully discrete method is presented for computing inertial manifolds of dissipative partial differential equations. In particular, only an approximate spectral decomposition of the dominant differential operator needs to be known. The first few of the smallest eigenvalues and eigenvectors of the discretized operator are approximated using the Lanczos algorithm. Numerical experiments are performed for an equation in one space dimension by discretizing the space variable on a sufficiently fine grid. The basic ideas and techniques are exemplified for selected bifurcation diagrams of an integrated form of the Kuramoto-Sivashinsky equation.This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) grant OGP0036901, NSERC and Schweizerischer Nationalfonds zur Förderung der Wissenschaften BEF 0150297, and Forschungsinstitut für Mathematik, ETH Zürich.