@Article{IJNAM-17-3,
author = {Öffner, Philipp and Jan, Glaubitz and Ranocha, Hendrik},
title = {Analysis of Artificial Dissipation of Explicit and Implicit Time-Integration Methods},
journal = {International Journal of Numerical Analysis and Modeling},
year = {2020},
volume = {17},
number = {3},
pages = {332--349},
abstract = {<p style="text-align: justify;">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) <strong>L</strong><sub>2</sub> 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.</p>},
issn = {2617-8710},
doi = {https://doi.org/2020-IJNAM-16862},
url = {https://global-sci.com/article/83022/analysis-of-artificial-dissipation-of-explicit-and-implicit-time-integration-methods}
}