Local Time-Stepping for Explicit Discontinuous Galerkin Schemes [chapter]

Gregor Gassner, Michael Dumbser, Florian Hindenlang, Claus-Dieter Munz
2011 Computational Fluid Dynamics 2010  
A class of explicit discontinuous Galerkin schemes is described which time approximation is based on a predictor corrector formulation. The approximation at the new time level is obtained in one step with use of the information from the direct neighbors only. This allows to introduce a local time-stepping for unsteady simulations with the property that every grid cell runs with its own optimal time step. Abstract The present paper deals with the continuous work of extending the multidimensional
more » ... limiting process (MLP), which has been quite successful in finite volume methods (FVM), into discontinuous Galerkin (DG) methods. Based on successful analyses and implementations of the MLP slope limiting in FVM, MLP is applicable into DG framework with the MLP-based troubled-cell marker and the MLP slope limiter. Through several test cases, it is observed that the newly developed MLP combined with DG methods provides quite desirable performances in controlling numerical oscillations as well as capturing key flow features. Abstract High-order compact difference scheme (CD) based on the half-staggered mesh is compared with discontinuous Galerkin method in computations of the incompressible flow. Assessment of the accuracy is performed based on the classical test cases: Taylor-Green vortices, Burggraf flow and also for temporally evolving shear layer. The CD method method provides very accurate results with expected order of accuracy, 4th and 6th. Similarly for the discontinuous Galerkin method provided that the number of degrees of fredom is close to the number of nodes in computations with CD method. Furthermore, it appeared that CD method is much more efficient than the discontinuous Galerkin method of comparable accuracy. Abstract An appropriate boundary treatment is one of the most important tasks to perform when carrying out numerical simulations. The technique to define the boundary condition depends strongly on both the numerical scheme and the type of differential equation to be solved. In terms of implementation effort and cost of computational resources, every boundary should be treated locally in both space and time. In this paper we discuss techniques to deal with adiabatic walls in the framework of high-order discontinuous Galerkin methods for compressible flow.
doi:10.1007/978-3-642-17884-9_19 fatcat:ydan2psyffc23othhziaotksxu