Discontinuous Galerkin methods: exploiting superconvergence for improved time-stepping

Frean, Daniel (2017) Discontinuous Galerkin methods: exploiting superconvergence for improved time-stepping. Doctoral thesis, University of East Anglia.

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    The discontinuous Galerkin (DG) methods are one of the most extensively researched classes of numerical methods for solving partial dfferential equations that display convective or diffusive qualities and have been popularly adopted by the scientific and engineering communities as a method capable of achieving arbitrary orders of accuracy in space. The choice of numerical flux function plays a pivotal role in the successful construction of DG methods and has an intrinsic effect on the superconvergence properties. As an inherent property of the spatial discretisation, superconvergence can only be retained in the solution through a sensitive pairing with a time integrator. The results of the literature and of this work suggest that an improved pairing between the spatial and temporal discretisations is both desirable and possible.

    We perform analysis of three different but related manifestations of superconvergence: the local, super-accurate points themselves; the subsequent global extraction via the Smoothness-Increasing Accuracy-Conserving (SIAC) filters; and the spectral properties that quantify, in terms of dispersion and dissipation errors, how accurately waves are convected. In order to explore the effect of the numerical flux function on superconvergence, we consider a generalisation of the “natural" upwind choice for a Method of Lines solution to the linear advection equation: the upwind-biased flux. We prove that the method is locally superconvergent at roots of a linear combination of the left- and right-Radau polynomials dependent on the value of a flux parameter and that the use of SIAC filters is still able to draw out the superconvergence information and create a globally smooth and superconvergent solution. In exploring the coupling of DG with a time integrator, we introduce a new scheme to a class of multi-stage multi-derivative methods, following recent incorporation of local DG technologies to recover superconvergence and achieve improved wave propagation properties.

    Item Type: Thesis (Doctoral)
    Faculty \ School: Faculty of Science > School of Mathematics
    Depositing User: Gillian Aldus
    Date Deposited: 22 Mar 2018 11:52
    Last Modified: 22 Mar 2018 11:52
    URI: https://ueaeprints.uea.ac.uk/id/eprint/66543

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