Geodesic
Accuracy comparison for discontinuous Galerkin schemes in the case of wave and vortex
Matematičeskoe modelirovanie, Tome 31 (2019) no. 10, pp. 22-34.

Voir la notice de l'article provenant de la source Math-Net.Ru

This study considers discontinuous Galerkin schemes with Legendre polynomial basis of degree $K=0,\dots,5$. The scheme build for convection equation is studied in sense of amplitude and dispersion error. The relation between wave number and cell size is determined to provide the desired solution accuracy in task with wave. The computations for acoustic wave are performed with the solver for full Euler equations. Despite different problem statement, the analytical estimates for the relation prove practical applicability. The other test considers inviscid vortex flow. The practical estimate of mesh density is based on the solution of model vortex flow with viscosity. The relation between scheme order and mesh density is pointed for all testcases. The results of this paper can be used in building meshes for tasks intended to be solved with discontinuous Galerkin approach.
Keywords: discontinuous Galerkin method, acoustic wave
Mots-clés : convection equation, Euler equations, vortex. 
@article{MM_2019_31_10_a2,
     author = {I. S. Bosnyakov},
     title = {Accuracy comparison for discontinuous {Galerkin} schemes in the case of wave and vortex},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {22--34},
     publisher = {mathdoc},
     volume = {31},
     number = {10},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2019_31_10_a2/}
}
TY  - JOUR
AU  - I. S. Bosnyakov
TI  - Accuracy comparison for discontinuous Galerkin schemes in the case of wave and vortex
JO  - Matematičeskoe modelirovanie
PY  - 2019
SP  - 22
EP  - 34
VL  - 31
IS  - 10
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MM_2019_31_10_a2/
LA  - ru
ID  - MM_2019_31_10_a2
ER  - 
%0 Journal Article
%A I. S. Bosnyakov
%T Accuracy comparison for discontinuous Galerkin schemes in the case of wave and vortex
%J Matematičeskoe modelirovanie
%D 2019
%P 22-34
%V 31
%N 10
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MM_2019_31_10_a2/
%G ru
%F MM_2019_31_10_a2
I. S. Bosnyakov. Accuracy comparison for discontinuous Galerkin schemes in the case of wave and vortex. Matematičeskoe modelirovanie, Tome 31 (2019) no. 10, pp. 22-34. http://geodesic.mathdoc.fr/item/MM_2019_31_10_a2/

[1] N. Kroll, C. Hirsch, F. Bassi, C. Johnston, K. Hillewaert (eds.), Industrialization of High-Order Methods-A Top-Down Approach: Results of a Collaborative Research Project Funded by the European Union 2010–2014, Springer, 2015 | MR | Zbl

[2] M. V. Lipavskii, A. D. Savel'ev, A. I. Tolstykh, D. A. Shirobokov, “Multioperator schemes up to the 18th order of accuracy with applications to problems of instability and jet acoustics”, TsAGI Science Journal, 43:3 (2012)

[3] R. C. Moura, S. J. Sherwin, J. Peiro, “Linear dispersion-diffusion analysis and its application to under-resolved turbulence simulations using discontinuous Galerkin spectral/hp methods”, J. Comput. Phys., 298 (2015), 659–710 | DOI | MR

[4] S. J. Sherwin, “Dispersion analysis of the continuous and discontinuous Galerkin formulations”, Discontinuous Galerkin Methods, Springer, Berlin–Heidelberg, 2000, 425–431 | DOI | MR | Zbl

[5] S. M. Bosnyakov, S. V. Mikhaylov V. Yu. Podaruev, A. I. Troshin, “Unsteady Discontinuous Galerkin Method of a High Order of Accuracy for Modeling Turbulent Flows”, Mathematical Models and Computer Simulations, 11:1 (2019), 22–34 | DOI | MR

[6] S. V. Mihaylov, A. N. Morozov, V. Yu. Podaruev, A. V. Wolkov, V. V. Vlasenko, “Verifikatsiia chislennoi skhemy razryvnogo metoda Galerkina primenitelno k linearizovannym uravneniiam Eulera”, Rezultaty fundamentalnykh issledovanii v prikladnykh zadachakh aviastroeniia, M., 2016, 228–239

[7] L. D. Landau, E. M. Lifshitz, Fluid Mechanics, v. 6, 2nd ed., Butterworth-Heinemann | MR | MR

[8] B. Cockburn, C. W. Shu, “The Runge-Kutta Discontinuous Galerkin Method for Conservation Laws V”, J. Comput. Phys, 141:2 (1998), 199–224 | DOI | MR | Zbl