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High Order Discontinuous Galerkin Methods For Systems Of Conservation Laws
==HIGH ORDER DISCONTINUOUS GALERKIN METHODS FOR SYSTEMS OF CONSERVATION LAWS== Jaime Peraire (Massachusetts Institute of Technology) The talk will describe our latest developments in Discontinuous Galerkin methods for solving systems of conservations laws. We will describe our low cost Compact Discontinuous Galerkin method for discretizing diffusive operators, solution techniques and shock capturing approaches. A more recent area of research is that of hybridizable discontinuous Galerkin methods. In these methods, the approximate variable and corresponding flux within each element are expressed in terms of an approximate trace of the original variable along the element boundary. A unique value for the approximate trace is obtianed by enforcing the continuity of the normal component of the flux across the element boundary and a global equation system solely in terms of the approximate trace is thus obtained. One of the advantages of these hybridizable methods is that they can produce superconvergent results for the viscous fluxes in multidimensions. That is, if an approximation of order p is used for the original variable, it is possible to obtain approximations for the derivative (or flux) of this variable which converges like p + 1. Finally, it is possible to introduce a simple element-by-element postprocessing scheme to obtain new approximations of the flux and the original variable. The new approximate flux, which has a continuous interelement normal component, is shown to converge with order p + 1 in the L2 −norm. The new approximate variable is shown to converge with order p + 2 in the L2 −norm. For the time dependent case, the postprocessing does not need to be applied at each time step but only at the times for which an enhanced solution is required. Applications involving fluid dynamics as well as nonlinear solid dynamics and fluid structure interaction will be presented.
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