Specialized numerical methods for time dependent partial differential equations

Prof. Donald French

Department of Mathematics, University of Cinncinatti and University of Maryland Baltimore County

Accurate representation of the qualitative aspects of partial differential equation (pde) problems is useful and often crucial in the calculation of a sensible numerical approximation. We look at several physical models where this approach has been successfully applied, highlighting important properties of the pde and the way in which the numerical method retains them. We will also discuss a more general approach called the continuous Galerkin (cG) method which is a technique to develop time discretizations using finite elements and is similar to the discontinuous Galerkin (dG) method. The cG method retains energy properties for pde models and, like the dG method, provides a framework for the derivation of a posteriori estimates (useful for the creation of adaptive schemes) and analysis of the accuracy of the approximation.