On the solution of linear algebraic systems arising from the semi–implicit DGFE discretization of the compressible Navier–Stokes equations

Vít Dolejší

Kybernetika (2010)

  • Volume: 46, Issue: 2, page 260-280
  • ISSN: 0023-5954

Abstract

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We deal with the numerical simulation of a motion of viscous compressible fluids. We discretize the governing Navier–Stokes equations by the backward difference formula – discontinuous Galerkin finite element (BDF-DGFE) method, which exhibits a sufficiently stable, efficient and accurate numerical scheme. The BDF-DGFE method requires a solution of one linear algebra system at each time step. In this paper, we deal with these linear algebra systems with the aid of an iterative solver. We discuss the choice of the preconditioner, stopping criterion and the choice of the time step and propose a new strategy which leads to an efficient and accurate numerical scheme.

How to cite

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Dolejší, Vít. "On the solution of linear algebraic systems arising from the semi–implicit DGFE discretization of the compressible Navier–Stokes equations." Kybernetika 46.2 (2010): 260-280. <http://eudml.org/doc/196789>.

@article{Dolejší2010,
abstract = {We deal with the numerical simulation of a motion of viscous compressible fluids. We discretize the governing Navier–Stokes equations by the backward difference formula – discontinuous Galerkin finite element (BDF-DGFE) method, which exhibits a sufficiently stable, efficient and accurate numerical scheme. The BDF-DGFE method requires a solution of one linear algebra system at each time step. In this paper, we deal with these linear algebra systems with the aid of an iterative solver. We discuss the choice of the preconditioner, stopping criterion and the choice of the time step and propose a new strategy which leads to an efficient and accurate numerical scheme.},
author = {Dolejší, Vít},
journal = {Kybernetika},
keywords = {discontinuous Galerkin method; compressible Navier–Stokes equations; linear algebra problems; preconditioning; stopping criterion; choice of the time step; discontinuous Galerkin method; compressible Navier-Stokes equations; linear algebra problems; preconditioning; stopping criterion; choice of the time step},
language = {eng},
number = {2},
pages = {260-280},
publisher = {Institute of Information Theory and Automation AS CR},
title = {On the solution of linear algebraic systems arising from the semi–implicit DGFE discretization of the compressible Navier–Stokes equations},
url = {http://eudml.org/doc/196789},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Dolejší, Vít
TI - On the solution of linear algebraic systems arising from the semi–implicit DGFE discretization of the compressible Navier–Stokes equations
JO - Kybernetika
PY - 2010
PB - Institute of Information Theory and Automation AS CR
VL - 46
IS - 2
SP - 260
EP - 280
AB - We deal with the numerical simulation of a motion of viscous compressible fluids. We discretize the governing Navier–Stokes equations by the backward difference formula – discontinuous Galerkin finite element (BDF-DGFE) method, which exhibits a sufficiently stable, efficient and accurate numerical scheme. The BDF-DGFE method requires a solution of one linear algebra system at each time step. In this paper, we deal with these linear algebra systems with the aid of an iterative solver. We discuss the choice of the preconditioner, stopping criterion and the choice of the time step and propose a new strategy which leads to an efficient and accurate numerical scheme.
LA - eng
KW - discontinuous Galerkin method; compressible Navier–Stokes equations; linear algebra problems; preconditioning; stopping criterion; choice of the time step; discontinuous Galerkin method; compressible Navier-Stokes equations; linear algebra problems; preconditioning; stopping criterion; choice of the time step
UR - http://eudml.org/doc/196789
ER -

References

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  1. Arnold, D. N., 10.1137/0719052, SIAM J. Numer. Anal. 19 (1982), 4, 742–760. Zbl0482.65060MR0664882DOI10.1137/0719052
  2. Arnold, D. N., Brezzi, F., Cockburn, B., Marini, L. D., 10.1137/S0036142901384162, SIAM J. Numer. Anal. 39 (2002), 5, 1749–1779. Zbl1008.65080MR1885715DOI10.1137/S0036142901384162
  3. Bassi, F., Rebay, S., 10.1006/jcph.1996.5572, J. Comput. Phys. 131 (1997), 267–279. Zbl0871.76040MR1433934DOI10.1006/jcph.1996.5572
  4. Bassi, F., Rebay, S., A high order discontinuous Galerkin method for compressible turbulent flow, In: Discontinuous Galerkin Method: Theory, Computations and Applications (B. Cockburn, G. E. Karniadakis, and C. W. Shu, eds.), (Lecture Notes in Computat. Sci. Engrg. 11.) Springer-Verlag, Berlin 2000, pp. 113–123. MR1842164
  5. Baumann, C. E., Oden, J. T., 10.1002/(SICI)1097-0363(19990915)31:1<79::AID-FLD956>3.0.CO;2-C, Internat. J. Numer. Methods Fluids 31 (1999), 1, 79–95. Zbl0985.76048MR1714511DOI10.1002/(SICI)1097-0363(19990915)31:1<79::AID-FLD956>3.0.CO;2-C
  6. Ciarlet, P. G., The Finite Elements Method for Elliptic Problems, North-Holland, Amsterdam – New York – Oxford 1979. MR0520174
  7. Cockburn, B., Hou, S., Shu, C. W., TVB Runge–Kutta local projection discontinuous Galerkin finite element for conservation laws IV: The multi-dimensional case, Math. Comp. 54 (1990), 545–581. MR1010597
  8. Dawson, C. N., Sun, S., Wheeler, M. F., 10.1016/j.cma.2003.12.059, Comput. Meth. Appl. Mech. Engrg. 193 (2004), 2565–2580. Zbl1067.76565MR2055253DOI10.1016/j.cma.2003.12.059
  9. Dolejší, V., 10.1002/fld.730, Internat. J. Numer. Methods Fluids 45 (2004), 1083–1106. MR2072224DOI10.1002/fld.730
  10. Dolejší, V., Semi-implicit interior penalty discontinuous Galerkin methods for viscous compressible flows, Commun. Comput. Phys. 4 (2008), 2, 231–274. MR2440946
  11. Dolejší, V., Kůs, P., 10.1002/nme.2143, Internat. J. Numer. Methods Engrg. 73 (2008), 12, 1739–1766. MR2397970DOI10.1002/nme.2143
  12. Dolejší, V., Discontinuous Galerkin method for the numerical simulation of unsteady compressible flow, WSEAS Trans. on Systems 5 (2006), 5, 1083–1090. 
  13. Dolejší, V., Feistauer, M., 10.1016/j.jcp.2004.01.023, J. Comput. Phys. 198 (2004), 2, 727–746. MR2062915DOI10.1016/j.jcp.2004.01.023
  14. Dumbser, M., Munz, C. D., 10.1007/s10915-005-9025-0, J. Sci. Comput. 27 (2006), 215–230. MR2285777DOI10.1007/s10915-005-9025-0
  15. Feistauer, M., Felcman, J., Straškraba, I., Mathematical and Computational Methods for Compressible Flow, Oxford University Press, Oxford 2003. MR2261900
  16. Feistauer, M., Kučera, V., 10.1016/j.jcp.2007.01.035, J. Comput. Phys. 224 (2007), 1, 208–221. MR2322268DOI10.1016/j.jcp.2007.01.035
  17. Feistauer, M., Kučera, V., Prokopová, J., 10.1016/j.matcom.2009.01.020, Math. Comput. Simulations 80 (2010), 8, 1612-1623. MR2647255DOI10.1016/j.matcom.2009.01.020
  18. Hairer, E., Norsett, S. P., Wanner, G., Solving ordinary differential equations I, Nonstiff problems, (Springer Series in Computational Mathematics No. 8.) Springer Verlag, Berlin 2000. MR1227985
  19. Hartmann, R., Houston, P., Symmetric interior penalty DG methods for the compressible Navier–Stokes equations I: Method formulation, Internat. J. Numer. Anal. Model. 1 (2006), 1–20. Zbl1129.76030MR2208562
  20. Klaij, C. M., Vegt, J. van der, Ven, H. V. der, 10.1016/j.jcp.2006.04.003, J. Comput. Phys. 219 (2006), 2, 622–643. MR2274951DOI10.1016/j.jcp.2006.04.003
  21. Lörcher, F., Gassner, G., Munz, C. D., 10.1007/s10915-007-9128-x, I. Inviscid compressible flow in one space dimension. J. Sci. Comput. 32 (2007), 2, 175–199. MR2320569DOI10.1007/s10915-007-9128-x
  22. Rivière, B., Wheeler, M. F., Girault, V., Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems, I. Comput. Geosci. 3 (1999), 3-4, 337–360. MR1750076
  23. Watkins, D. S., Fundamentals of Matrix Computations, (Pure and Applied Mathematics, Wiley-Interscience Series of Texts, Monographs, and Tracts.) John Wiley , New York 2002. MR1899577

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