@article{37951,
  keywords = {inertial manifolds, Kuramoto-Sivashinsky equation},
  author = {FOIAS and Jolly and Kevrekidis and ES TITI},
  title = {DISSIPATIVITY OF NUMERICAL SCHEMES},
  abstract = { We show that the way in which finite differences are applied to the nonlinear term in certain partial differential equations (PDES) can mean the difference between dissipation and blow up. For fixed parameter values and arbitrarily fine discretizations we construct solutions which blow up in finite time for two semi-discrete schemes. We also show the existence of spurious steady states whose unstable manifolds, in some cases, contain solutions which explode. This connection between the blow-up phenomenon and spurious steady states is also explored for Galerkin and nonlinear Galerkin semi-discrete approximations. Two fully discrete finite difference schemes derived from a third semi-discrete scheme, reported by Foias and Titi to be dissipative, are analysed. Both latter schemes are shown to have a stability condition which is independent of the initial data. A similar result is obtained for a fully discrete Galerkin scheme. While the results are stated for the Kuramoto-Sivashinsky equation, most naturally carry over to other dissipative PDES. },
  year = {1991},
  journal = {NonlinearityNonlinearity},
  volume = {4},
  pages = {591-613},
  month = {08/1991},
  isbn = {0951-7715},
  language = {English},
}