 
Summary: Discontinuous Galerkin finite element
methods for hyperbolic nonconservative
partial differential equations
S. Rhebergen , O. Bokhove and J.J.W. van der Vegt
Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500
AE, Enschede, The Netherlands
Abstract
We present space and spacetime discontinuous Galerkin finite element (DGFEM)
formulations for systems containing nonconservative products, such as occur in dis
persed multiphase flow equations. The main criterium we pose on the weak formu
lation is that if the system of nonconservative partial differential equations can be
transformed into conservative form, then the formulation must reduce to that for
conservative systems. Standard DGFEM formulations cannot be applied to noncon
servative systems of partial differential equations. We therefore introduce the theory
of weak solutions for nonconservative products into the DGFEM formulation leading
to the new question how to define the path connecting left and right states across a
discontinuity. The effect of different paths on the numerical solution is investigated
and found to be small. We also introduce a new numerical flux that is able to deal
with nonconservative products. Our scheme is applied to two different systems of
partial differential equations. First, we consider the shallow water equations, where
