The Discontinuous Galerkin Method for Conservation Laws

Michael, Crabb (2010) The Discontinuous Galerkin Method for Conservation Laws. Masters thesis, The University of Manchester.

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The aim of this project is to study discontinuous Galerkin methods applied to coupled systems of partial differential equations in conservative form in 1D and 2D. In 1D, a formulation was successfully implemented to solve continuous problems for the advection and shallow water equations. Discontinuous problems for the inviscid Burgers� equation and a breaking dam problem were also investigated and the effectiveness of h- and p-refinement discussed. An alternate set of shallow water equations were derived yielding equivalent results for a continuous problem but different numerical solutions for the breaking dam problem. These anomalous results highlight the importance of enforcing conservation of the correct conserved physical variables in cases when solutions exhibit shocks. A 2D slope limiter, applicable to quadrilateral elements, is implemented and numerical results obtained for a smoothed breaking dam problem in 2D. A comparison is made between these results and those from a finite volume method (results by Chris Johnson) and indicate that, for this particular problem, both methods resolve the shock over the same length scale.

Item Type: Thesis (Masters)
Uncontrolled Keywords: Finite Elements, Discontinuous Galerkin, Conservation Laws, Slope Limiter
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Dr Michael Crabb
Date Deposited: 27 Jul 2015
Last Modified: 20 Oct 2017 14:13

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