On discontinuous Galerkin methods for nonlinear convection-diffusion problems and compressible flow

Vít Dolejší; Miloslav Feistauer; Christoph Schwab

Mathematica Bohemica (2002)

  • Volume: 127, Issue: 2, page 163-179
  • ISSN: 0862-7959

Abstract

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The paper is concerned with the discontinuous Galerkin finite element method for the numerical solution of nonlinear conservation laws and nonlinear convection-diffusion problems with emphasis on applications to the simulation of compressible flows. We discuss two versions of this method: (a) Finite volume discontinuous Galerkin method, which is a generalization of the combined finite volume—finite element method. Its advantage is the use of only one mesh (in contrast to the combined finite volume—finite element schemes). However, it is of the first order only. (b) Pure discontinuous Galerkin finite element method of higher order combined with a technique avoiding spurious oscillations in the vicinity of shock waves.

How to cite

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Dolejší, Vít, Feistauer, Miloslav, and Schwab, Christoph. "On discontinuous Galerkin methods for nonlinear convection-diffusion problems and compressible flow." Mathematica Bohemica 127.2 (2002): 163-179. <http://eudml.org/doc/249044>.

@article{Dolejší2002,
abstract = {The paper is concerned with the discontinuous Galerkin finite element method for the numerical solution of nonlinear conservation laws and nonlinear convection-diffusion problems with emphasis on applications to the simulation of compressible flows. We discuss two versions of this method: (a) Finite volume discontinuous Galerkin method, which is a generalization of the combined finite volume—finite element method. Its advantage is the use of only one mesh (in contrast to the combined finite volume—finite element schemes). However, it is of the first order only. (b) Pure discontinuous Galerkin finite element method of higher order combined with a technique avoiding spurious oscillations in the vicinity of shock waves.},
author = {Dolejší, Vít, Feistauer, Miloslav, Schwab, Christoph},
journal = {Mathematica Bohemica},
keywords = {discontinuous Galerkin finite element method; numerical flux; conservation laws; convection-diffusion problems; limiting of order of accuracy; numerical solution of compressible Euler equations; discontinuous Galerkin finite element method; numerical flux; conservation laws; convection-diffusion problems; limiting of order of accuracy; numerical solution of compressible Euler equations},
language = {eng},
number = {2},
pages = {163-179},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On discontinuous Galerkin methods for nonlinear convection-diffusion problems and compressible flow},
url = {http://eudml.org/doc/249044},
volume = {127},
year = {2002},
}

TY - JOUR
AU - Dolejší, Vít
AU - Feistauer, Miloslav
AU - Schwab, Christoph
TI - On discontinuous Galerkin methods for nonlinear convection-diffusion problems and compressible flow
JO - Mathematica Bohemica
PY - 2002
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 127
IS - 2
SP - 163
EP - 179
AB - The paper is concerned with the discontinuous Galerkin finite element method for the numerical solution of nonlinear conservation laws and nonlinear convection-diffusion problems with emphasis on applications to the simulation of compressible flows. We discuss two versions of this method: (a) Finite volume discontinuous Galerkin method, which is a generalization of the combined finite volume—finite element method. Its advantage is the use of only one mesh (in contrast to the combined finite volume—finite element schemes). However, it is of the first order only. (b) Pure discontinuous Galerkin finite element method of higher order combined with a technique avoiding spurious oscillations in the vicinity of shock waves.
LA - eng
KW - discontinuous Galerkin finite element method; numerical flux; conservation laws; convection-diffusion problems; limiting of order of accuracy; numerical solution of compressible Euler equations; discontinuous Galerkin finite element method; numerical flux; conservation laws; convection-diffusion problems; limiting of order of accuracy; numerical solution of compressible Euler equations
UR - http://eudml.org/doc/249044
ER -

References

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  1. 10.1023/A:1023217905340, Appl. Math. 43 (1998), 263–310. (1998) MR1627989DOI10.1023/A:1023217905340
  2. 10.1006/jcph.1996.5572, J. Comput. Phys. 131 (1997), 267–279. (1997) MR1433934DOI10.1006/jcph.1996.5572
  3. Discontinuous Galerkin methods for convection dominated problems, High-Order Methods for Computational Physics, T. J. Barth, H. Deconinck (eds.), Lecture Notes in Computational Science and Engineering 9, Springer, Berlin, 1999, pp. 69–224. (1999) Zbl0937.76049MR1712278
  4. Discontinuous Galerkin Methods, Lect. Notes Comput. Sci. Eng. 11., Springer, Berlin, 2000. (2000) MR1842160
  5. 10.1007/s007910050015, Comput. Vis. Sci. 1 (1998), 165–178. (1998) DOI10.1007/s007910050015
  6. A finite volume discontinuous Galerkin scheme for nonlinear convection-diffusion problems, Calcolo. Preprint, Forschungsinstitut für Mathematik ETH Zürich, Januar 2001 (to appear). (to appear) MR1901200
  7. On some aspects of the discontinuous Galerkin finite element method for conservation laws, Mathematics and Computers in Simulation. The Preprint Series of the School of Mathematics, Charles University Prague, No. MATH-KNM-2001/5, 2001 (to appear). (to appear) MR1984135
  8. Mathematical Methods in Fluid Dynamics, Longman Scientific & Technical, Harlow, 1993. (1993) Zbl0819.76001
  9. Numerical methods for compressible flow, Mathematical Fluid Mechanics. Recent Results and Open Questions, J. Neustupa, P. Penel (eds.), Birkhäuser, Basel, 2001, pp. 105–142. Zbl1036.76035MR1865051
  10. Theory and applications of numerical schemes for nonlinear convection-diffusion problems and compressible Navier-Stokes equations, The Mathematics of Finite Elements and Applications, J. R. Whiteman (ed.), Highlights, 1996, Wiley, Chichester, 1997, pp. 175–194. 
  11. Numerical solution of compressible flow through cascades of profiles, Z. Angew. Math. Mech. 76 (1996), 297–300. (1996) 
  12. 10.1002/(SICI)1098-2426(199703)13:2<163::AID-NUM3>3.0.CO;2-N, Numer. Methods Partial Differ. Equations 13 (1997), 163–190. (1997) MR1436613DOI10.1002/(SICI)1098-2426(199703)13:2<163::AID-NUM3>3.0.CO;2-N
  13. 10.1137/S0036142997314695, SIAM J. Numer. Anal. 36 (1999), 1528–1548. (1999) MR1706727DOI10.1137/S0036142997314695
  14. 10.1002/(SICI)1098-2426(199903)15:2<215::AID-NUM6>3.0.CO;2-1, Explicit schemes. Numer. Methods Partial Differ. Equations 15 (1999), 215–235. (1999) MR1674294DOI10.1002/(SICI)1098-2426(199903)15:2<215::AID-NUM6>3.0.CO;2-1
  15. On a 3D adaptation for compressible flow, Proceedings of the Conf. Finite Element Methods for Three-Dimensional Problems, Jyväskylä, June 27–July 1, 2000 (to appear). (to appear) Zbl0996.76060
  16. Numerical solution of several 2D and 3D internal and external flow problems, Numerical Modelling in Continuum Mechanics, M. Feistauer, R. Rannacher, K. Kozel (eds.), Matfyzpress, Praha, 1997, pp. 283–291. 
  17. Adaptive discontinuous Galerkin finite element methods for nonlinear conservation laws, Preprint 2001–20, Mai 2001, SFB 359, Universität Heidelberg. 
  18. 10.1006/jcph.1998.6032, J. Comput. Phys. 146 (1998), 491–519. (1998) MR1654911DOI10.1006/jcph.1998.6032
  19. Multigrid Solution of the Steady Euler Equations, Centrum voor Wiskunde en Informatica, Amsterdam, 1987. (1987) MR0942891

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