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Phone: (407) 823-6284;   Fax: (407) 823-6253;   MAP  207 2004-2005 MATHEMATICS COLLOQUIUM SERIES 01/18/05 Colloquium Dr. Bengt Fornberg Department of Applied Mathematics University of Colorado, Boulder Pseudospectral Methods For High Order Initial-Boundary Value Problems Abstract:   When pseudospectral approximations are used for space derivatives, one often encounters spurious eigenvalues. These can lead to severe time stepping difficulties for PDEs. This is especially the case for equations with high order derivatives in space, requiring multiple conditions at one or both boundaries. We note here that a very simple-to-implement fictitious point approach circumvents most of these difficulties. The new approach is tested on the Kuramoto-Sivashinsky equation. We conclude by noting that highly accurate solution methods for initial-boundary value problems makes time-space corner singularities for such problems an issue of increasing importance.