Geodesic
Computational efficiency of ADER and RK schemes for discontinuous Galerkin method in case of 1D Hopf equation
Matematičeskoe modelirovanie, Tome 33 (2021) no. 7, pp. 109-120.

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

This paper considers Discontinuous Galerkin schemes based on Legendre polynomials of degree $K=2, 3$. Schemes are written to solve the one-dimensional Hopf equation. Unsteady solution is acquired with ADER and Runge-Kutta algorithms. The high order of numerical approaches is affirmed. The ADER method computational efficiency is studied in comparison with traditional approach. Tests that are used are with an analytical solution (linear solution and running half-wave), and with Burgers turbulence. The result of this work can be used to speed up 3D DG-based algorithms.
Keywords: discontinuous Galerkin method, Hopf equation, efficiency, ADER, Runge–Kutta, high-order.
Mots-clés : burgulence 
@article{MM_2021_33_7_a8,
     author = {I. S. Bosnyakov and N. A. Klyuev},
     title = {Computational efficiency of {ADER} and {RK} schemes for discontinuous {Galerkin} method in case of {1D} {Hopf} equation},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {109--120},
     publisher = {mathdoc},
     volume = {33},
     number = {7},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2021_33_7_a8/}
}
TY  - JOUR
AU  - I. S. Bosnyakov
AU  - N. A. Klyuev
TI  - Computational efficiency of ADER and RK schemes for discontinuous Galerkin method in case of 1D Hopf equation
JO  - Matematičeskoe modelirovanie
PY  - 2021
SP  - 109
EP  - 120
VL  - 33
IS  - 7
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MM_2021_33_7_a8/
LA  - ru
ID  - MM_2021_33_7_a8
ER  - 
%0 Journal Article
%A I. S. Bosnyakov
%A N. A. Klyuev
%T Computational efficiency of ADER and RK schemes for discontinuous Galerkin method in case of 1D Hopf equation
%J Matematičeskoe modelirovanie
%D 2021
%P 109-120
%V 33
%N 7
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MM_2021_33_7_a8/
%G ru
%F MM_2021_33_7_a8
I. S. Bosnyakov; N. A. Klyuev. Computational efficiency of ADER and RK schemes for discontinuous Galerkin method in case of 1D Hopf equation. Matematičeskoe modelirovanie, Tome 33 (2021) no. 7, pp. 109-120. http://geodesic.mathdoc.fr/item/MM_2021_33_7_a8/

[1] A. V. Wolkov, Ch. Hirsch, N. B. Petrovskaya, “Application of a higher order discontinuous Galerkin method in computational aerodynamics”, Math. Model. Nat Phenom., 6:3 (2011), 237–263 | DOI | MR | Zbl | Zbl

[2] C. Hirsch et al. (eds.), TILDA: Towards Industrial LES/DNS in Aeronautics, Notes on Numerical Fluid Mechanics and Multidisciplinary Design, Springer, 2021 | MR

[3] F. Bassi, L. Botti, A. Colombo, A. Ghidoni, F. Massa, “Linearly implicit Rosenbrock-type Runge-Kutta schemes applied to the discontinuous Galerkin solution of compressible and incompressible unsteady flows”, J. Computers Fluids, 118 (2015), 305–320 | DOI | MR | Zbl

[4] E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer, 2009, 738 | MR

[5] V. A. Titarev, E. F. Toro, “TVD fluxes for the high-order ADERS schemes”, J. of Scientific Computing, 24:3 (2005) | MR | Zbl

[6] I. S. Men'shov, “Increasing the order of approximation of Godunov-s scheme us-ing the generalized Riemann problem”, Comp. Math. Math Phys., 30:5 (1990), 54–65 | DOI | Zbl

[7] A. G. Kulikovskii, N. V. Pogorelov, A. Yu. Semenov, Matematicheskie voprosy chislennogo resheniia giperbolicheskikh system uravnenii, Fizmatlit, M., 2001, 608 pp. | MR

[8] M. Dumbser, C. D. Munz, “Building blocks for arbitrary high order discontinuous Galerkin schemes”, J. of Scientific Computing, 27:1-3 (2006), 215–230 | DOI | MR | Zbl

[9] R. C. Moura, S. J. Sherwin, J. Peiró, “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

[10] V. A. Titarev, E. F. Toro, “ADER: Arbitrary high order Godunov approach”, J. Sci. Comput., 17 (2002), 609–618 | DOI | MR | Zbl

[11] N. A. Klyuev, Issledovanie vychislitelnoi effectivnosti skhem integrirovaniia po vremeni dlia razryvnogo metoda Galyorkina, bakalavrskaia rabota, FAKT MFTI, Zhukovsky, 2020