INTEGRATED SPACE-TIME ADAPTIVE HP-REFINEMENT METHODS FOR PARABOLIC-SYSTEMS

We describe adaptive integrated space-time hp-refinement algorithms for one-dimensional vector systems of parabolic partial differential equations. Solutions are calculated using Galerkin's method with a piecewise polynomial hierarchical basis in space and singly implicit Runge-Kutta (SIRK) methods in time. A posteriori estimates of local spatial and temporal discretization errors are used with a priori error estimates to control spatial and temporal enrichment. New techniques simplify spatial error estimation with high-order approximation; integrate spatial and temporal discretization and enrichment; and enable the selection of future meshes and acceptance of partial time steps. A base hp-refinement strategy and several variants are developed and compared using a number of linear and nonlinear examples.