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Volume 17, Issue 3
Analysis of Artificial Dissipation of Explicit and Implicit Time-Integration Methods

Philipp Öffner, Jan Glaubitz & Hendrik Ranocha

Int. J. Numer. Anal. Mod., 17 (2020), pp. 332-349.

Published online: 2020-05

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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.

  • AMS Subject Headings

65M12, 65M20

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

philipp.oeffner@math.uzh.ch (Philipp Öffner)

j.glaubitz@tu-bs.de (Jan Glaubitz)

hendrik.ranocha@kaust.edu.sa (Hendrik Ranocha)

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@Article{IJNAM-17-332, author = {Öffner , PhilippGlaubitz , Jan 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 = {

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.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/16862.html} }
TY - JOUR T1 - Analysis of Artificial Dissipation of Explicit and Implicit Time-Integration Methods AU - Öffner , Philipp AU - Glaubitz , Jan AU - Ranocha , Hendrik JO - International Journal of Numerical Analysis and Modeling VL - 3 SP - 332 EP - 349 PY - 2020 DA - 2020/05 SN - 17 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/16862.html KW - Hyperbolic conservation laws, flux reconstruction, summation-by-parts, artificial viscosity, full discrete stability, time integration methods. AB -

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.

Philipp Öffner , Jan Glaubitz & Hendrik Ranocha . (2020). Analysis of Artificial Dissipation of Explicit and Implicit Time-Integration Methods. International Journal of Numerical Analysis and Modeling. 17 (3). 332-349. doi:
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