Theoretical and numerical studies of the P N P M DG schemes in one space dimension

Abdulatif Badenjki; Gerald G. Warnecke

Applications of Mathematics (2019)

  • Volume: 64, Issue: 6, page 599-635
  • ISSN: 0862-7940

Abstract

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We give a proof of the existence of a solution of reconstruction operators used in the P N P M DG schemes in one space dimension. Some properties and error estimates of the projection and reconstruction operators are presented. Then, by applying the P N P M DG schemes to the linear advection equation, we study their stability obtaining maximal limits of the Courant numbers for several P N P M DG schemes mostly experimentally. A numerical study explains how the stencils used in the reconstruction affect the efficiency of the schemes.

How to cite

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Badenjki, Abdulatif, and Warnecke, Gerald G.. "Theoretical and numerical studies of the $P_NP_M$ DG schemes in one space dimension." Applications of Mathematics 64.6 (2019): 599-635. <http://eudml.org/doc/294475>.

@article{Badenjki2019,
abstract = {We give a proof of the existence of a solution of reconstruction operators used in the $P_NP_M$ DG schemes in one space dimension. Some properties and error estimates of the projection and reconstruction operators are presented. Then, by applying the $P_NP_M$ DG schemes to the linear advection equation, we study their stability obtaining maximal limits of the Courant numbers for several $P_NP_M$ DG schemes mostly experimentally. A numerical study explains how the stencils used in the reconstruction affect the efficiency of the schemes.},
author = {Badenjki, Abdulatif, Warnecke, Gerald G.},
journal = {Applications of Mathematics},
keywords = {$P_NP_M$ DG scheme; piecewise polynomial; projection; reconstruction; least square; local continuous space time Galerkin method; discontinuous Galerkin; advection equation; conservation law; von Neumann stability analysis; time discretization},
language = {eng},
number = {6},
pages = {599-635},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Theoretical and numerical studies of the $P_NP_M$ DG schemes in one space dimension},
url = {http://eudml.org/doc/294475},
volume = {64},
year = {2019},
}

TY - JOUR
AU - Badenjki, Abdulatif
AU - Warnecke, Gerald G.
TI - Theoretical and numerical studies of the $P_NP_M$ DG schemes in one space dimension
JO - Applications of Mathematics
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 6
SP - 599
EP - 635
AB - We give a proof of the existence of a solution of reconstruction operators used in the $P_NP_M$ DG schemes in one space dimension. Some properties and error estimates of the projection and reconstruction operators are presented. Then, by applying the $P_NP_M$ DG schemes to the linear advection equation, we study their stability obtaining maximal limits of the Courant numbers for several $P_NP_M$ DG schemes mostly experimentally. A numerical study explains how the stencils used in the reconstruction affect the efficiency of the schemes.
LA - eng
KW - $P_NP_M$ DG scheme; piecewise polynomial; projection; reconstruction; least square; local continuous space time Galerkin method; discontinuous Galerkin; advection equation; conservation law; von Neumann stability analysis; time discretization
UR - http://eudml.org/doc/294475
ER -

References

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  9. Koornwinder, T. H., Wong, R., Koekoek, R., Swarttouw, R. F., Orthogonal polynomials, NIST Handbook of Mathematical Functions F. W. J. Olver et al. Cambridge University Press, Cambridge (2010), 435-484. (2010) Zbl1198.00002MR2655358
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  11. Strang, G., Introduction to Linear Algebra, Wellesley-Cambridge Press, Wellesley (2003). (2003) Zbl1046.15001MR3058665

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