New approaches to the construction of efficient high order time integration methods in the context of DG space discretisations for viscous and inviscid fluid flow

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Discontinuous space discretizations, especially the Discontinuous Galerkin(DG) methods, are a modern and popular class of numerical methods especially for computationally intensive fluid dynamics calculations. Their popularity is due to the fact that DG methods allow for high order approximations in combination with high flexibility - e.g. in choosing different polynomial degrees on neighbouring elements. Furthermore, the challenge of the future in order to enable reliable simulations of complex real life problems is the design of methods for parallel applications. Here. DG methods are perfectly suitable and thus it is necessary to develop and analyse them especially with respect to the time integration employed. In the context of practically relevant problems, semi-discrete DG equations are often extremely stiff. In the case of complex geometries, e.g. for fluid flow around obstacles, the DG mesh is locally refined with elements very different in size. In addition, for high Reynods numbers, applications require a considerable grid refinement in boundary layer zones. In this context, the time integration methods applied so far are yet far from being efficient. Especially with regard to the skillful coupling of explicit and implicit methods, as well as the use of local time steps as in multirate strategies, considerably more research is needed. In preliminary work, a robust, high order DG scheme with low numerical dissipation based on novel efficient filtering strategies has been developed. Based on this groundwork, the innovative contribution of this proposal is the development and analysis of novel IMEX time integration methods. In particular, for the first time we will incorporate hybrid approaches of the basic IMEX splitting combined with multirate methods in order to accelerate time integration of the semi-discrete DG equations. The main objective of this project is hence the development, analysis and the direct comparison of novel approaches to the construction of efficient, high order time integration schemes for viscous and inviscid fluid flow. These approaches will be studied in a uniform framework in order to develop suitable strategies to decide between IMEX or multirate method or to use a combination of both of them. In this context, stiffness detectors will be developed and analysed, we will assess concrete methods of implicit type within the IMEX approach and include multirate approaches as well. A further objective is to establish an analogy to IMEX and multirate approaches for exponential integrators which currently show considerable gain in efficiency. The The efficient time integration methods based on IMEX and multirate strategies which will be developed in this project will be highly suitable for practical applications. Hence they can be expected to set new standards both for the numerical calculation of fluid flow as for the simulation of phenomena based on fluid-structure-interaction which will be focussed on in the future.