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    Analysis of artificial dissipation of explicit and implicit time-integration methods

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    Type
    Article
    Authors
    Öffner, Philipp
    Glaubitz, Jan
    Ranocha, Hendrik cc
    KAUST Department
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Date
    2020-01-01
    Permanent link to this record
    http://hdl.handle.net/10754/665254
    
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    Abstract
    Stability is an important aspect of numerical methods for hyperbolic conservation laws and has received much interest. However, continuity in time is often assumed and only semidiscrete stability is studied. Thus, it is interesting to investigate the influence of explicit and implicit time integration methods on the stability of numerical schemes. If an explicit time integration method is applied, spacially stable numerical schemes for hyperbolic conservation laws can result in unstable fully discrete schemes. Focusing on the explicit Euler method (and convex combinations thereof), undesired terms in the energy balance trigger this phenomenon and introduce an erroneous growth of the energy over time. In this work, we study the influence of artificial dissipation and modal filtering in the context of discontinuous spectral element methods to remedy these issues. In particular, lower bounds on the strength of both artificial dissipation and modal filtering operators are given and an adaptive procedure to conserve the (discrete) L2 norm of the numerical solution in time is derived. This might be beneficial in regions where the solution is smooth and for long time simulations. Moreover, this approach is used to study the connections between explicit and implicit time integration methods and the associated energy production. By adjusting the adaptive procedure, we demonstrate that filtering in explicit time integration methods is able to mimic the dissipative behavior inherent in implicit time integration methods. This contribution leads to a better understanding of existing algorithms and numerical techniques, in particular the application of artificial dissipation as well as modal filtering in the context of numerical methods for hyperbolic conservation laws together with the selection of explicit or implicit time integration methods.
    Sponsors
    Jan Glaubitz was supported by the German Research Foundation (DFG, Deutsche Forschungsgemeinschaft) under Grant SO 363/15-1. Philipp Öffner was supported by SNF project\Solving advection dominated problems with high order schemes with polygonal meshes: Application to compressible and incompressible flow problems" and the UZH Postdoc Grant. Hendrik Ranocha was supported by the German Research Foundation (DFG, Deutsche Forschungsgemeinschaft) under Grant SO 363/14-1.
    Publisher
    University of Alberta
    Global Science Press
    Journal
    International Journal of Numerical Analysis and Modeling
    Additional Links
    http://www.global-sci.com/intro/article_detail/ijnam/16862.html
    http://www.math.ualberta.ca/ijnam/Volume-17-2020/No-3-20/2020-03-03.pdf
    Collections
    Articles; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

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