Novel implicit unconditionally stable time-stepping for DG-type methods and related topics
Aachen (2018) [Dissertation / PhD Thesis]
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The thesis is concerned with the improvement and evaluation of hybridized discontinuous Galerkin (HDG) methods for problems from computational fluid dynamics (CFD). HDG methods and high order methods in general promise to improve the quality of simulations used in research and development due to their high accuracy. However, due to their relatively young age there are open question regarding their stability, efficiency and applicability to a wide range of different problems. This thesis focuses on tackling some of these points. The main concern of this thesis are time-dependent, viscous flow problems that are often described by Euler or Navier-Stokes equations. Therefore, we investigate different types of implicit time integration methods of high order that have not been applied earlier to the HDG methods. This includes implicit general linear methods and multiderivative time integrators. The multiderivative time integrators introduce additional time derivatives that need to be approximated. We find that the approach used for explicit time integrators fails to deliver an uniformly stable discretization. Thus, we devise a new approach for implicit multiderivative integrators that remedies the stability issues. The efficient implementation of the numerical method is important to limit the required run-time of simulations. During this thesis we have developed several implementations which led to the contribution of an implementation of an HDG method to FESTUNG, an open source framework for MATLAB / GNU Octave. Besides efficiency we focus on a code that is well documented, extendable and easy to use. Many equations used for CFD are nonlinear and thus their solutions are prone to develop shocks. In order to handle shocks properly with the HDG method, a shock capturing method must be employed. We identify a suitable shock capturing method and justify our decision by comparing the method to other approaches. Additionally, we show that our choice allows to approximate several demanding test cases that contain shocks. The approximation of a solution to the equations we study requires to solve linear systems of equations. The solving process is a very time and memory consuming part of a simulation. Thus, it is necessary to have an efficient solution strategy. We discuss a newly introduced linear solver that makes use of the special structure of discontinuous Galerkin methods. We apply the new solver to nonlinear problems, which has not been done before, and compare it to an established linear solver to find good agreements in the results of both solvers.