Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations
Rhebergen, S. and Bokhove, O. and Vegt, J.J.W. van der (2007) Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations. [Report]

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Abstract:  We present space and spacetime discontinuous Galerkin finite element (DGFEM) formulations for systems containing nonconservative products, such as occur in dispersed multiphase flow equations. The main criterium we pose on the formulation is that if the system of nonconservative partial differential equations can be transformed into conservative form, then the formulation must reduce to that forconservative systems. Standard DGFEM formulations cannot be applied to nonconservative 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 topography leads to nonconservative products, in which the known, possibly discontinuous, topography is formally taken as an unknown in the system. Second, we consider a simplification of a depthaveraged twophase flow model which contains more intrinsic nonconservative products. 
Item Type:  Report 
Faculty:  Electrical Engineering, Mathematics and Computer Science (EEMCS) 
Research Group:  
Link to this item:  http://purl.utwente.nl/publications/66740 
Official URL:  http://www.math.utwente.nl/publications 
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