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Convergence of an implicit runge-kutta discontinuous Galerkin method using smooth limiters

Middag, Jacob (2012) Convergence of an implicit runge-kutta discontinuous Galerkin method using smooth limiters.

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Abstract:In higher order discontinuous Galerkin methods limiters are used to remove non-physical numerical oscillations. e limiters are used as a postprocessing step aer each stage or time step. However, the limited solution is not a solution to the DG formulation and oen limiters contain switches which are non-smooth. is results into a limit cycle behavior which hampers convergence of iterative methods used for the solution of algebraic equations resulting from an implicit time integration method. We will investigate a new smooth limiter in this thesis, the Weighted Biased Averaging Procedure (WBAP), to address this problem. e limiter is adapted to be used in an implicit discontinuous Galerkin method and the modifications necessary for a DG algorithm will be discussed in detail. e WBAP limiter is applied both in a one dimensional and a two dimensional seing. In the one dimensional seing this is done using shock tube problems described by the Euler equations for gas dynamics and steady state problems described by the Burgers equation. For the two dimensional problem we consider the shallow water equations for the flow in a channel with contraction. e application of the DG method is successful in both dimensions. However, the result of the simulations do not confirm the hypothesis that smoothness is beneficial for the convergence of the implicit time integration method. e most discontinuous WBAP variant is only slightly beer than the discontinuous minmod-TVB limiter and the other variants are worse. Compared to the results without limiter there is still a large gap in convergence rate. For future work it is recommended to look for methods that can deal with the discontinuous properties of limiters such as semi-smooth Newton methods.
Item Type:Essay (Master)
Faculty:EEMCS: Electrical Engineering, Mathematics and Computer Science
Subject:54 computer science
Programme:Computer Science MSc (60300)
Link to this item:http://purl.utwente.nl/essays/62122
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