Geodesic
On correlation between the discontinuous Galerkin method and MUSCL-type schemes
Matematičeskoe modelirovanie, Tome 27 (2015) no. 10, pp. 96-116.

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

The discontinuous Galerkin method is compared with MUSCL-type schemes. Description and analysis of schemes are presented as applied to the linear constant-coefficient advection equation. The techniques for generalizing schemes to nonlinear and multidimensional problems are considered. The distinctive features and correlations between the discontinuous Galerkin method and MUSCL-type schemes are revealed. The accuracy and efficiency criteria of the schemes when applied to different-type problems are discussed.
Keywords: Godunov-type schemes, Riemann problem, piecewise-linear approximation, Shu and Osher test problem, double Mach reflection test problem. 
@article{MM_2015_27_10_a7,
     author = {A. V. Rodionov},
     title = {On correlation between the discontinuous {Galerkin} method and {MUSCL-type} schemes},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {96--116},
     publisher = {mathdoc},
     volume = {27},
     number = {10},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2015_27_10_a7/}
}
TY  - JOUR
AU  - A. V. Rodionov
TI  - On correlation between the discontinuous Galerkin method and MUSCL-type schemes
JO  - Matematičeskoe modelirovanie
PY  - 2015
SP  - 96
EP  - 116
VL  - 27
IS  - 10
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MM_2015_27_10_a7/
LA  - ru
ID  - MM_2015_27_10_a7
ER  - 
%0 Journal Article
%A A. V. Rodionov
%T On correlation between the discontinuous Galerkin method and MUSCL-type schemes
%J Matematičeskoe modelirovanie
%D 2015
%P 96-116
%V 27
%N 10
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MM_2015_27_10_a7/
%G ru
%F MM_2015_27_10_a7
A. V. Rodionov. On correlation between the discontinuous Galerkin method and MUSCL-type schemes. Matematičeskoe modelirovanie, Tome 27 (2015) no. 10, pp. 96-116. http://geodesic.mathdoc.fr/item/MM_2015_27_10_a7/

[1] B. Cockburn, G. E. Karniadakis, C.-W. Shu, “The development of discontinuous Galerkin methods”, Discontinuous Galerkin Methods. Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, 11, Springer-Verlag GmbH, 2000, 3–50 | DOI | MR

[2] B. Cockburn, C.-W. Shu, “Runge–Kutta discontinuous Galerkin methods for convection-dominated problems”, J. Sci. Comput., 16 (2001), 173–261 | DOI | MR | Zbl

[3] M. Dumbser, “Building blocks for arbitrary high order discontinuous Galerkin schemes”, J. Sci. Comput., 27 (2006), 215–230 | DOI | MR | Zbl

[4] L. Krivodonova, “Limiters for high-order discontinuous Galerkin methods”, J. Comput. Phys., 226 (2007), 879–896 | DOI | MR | Zbl

[5] X. Zhong, C.-W. Shu, “A simple weighted nonoscillatory limiter for Runge–Kutta discontinuous Galerkin methods”, J. Comput. Phys., 232 (2013), 397–415 | DOI | MR

[6] M. E. Ladonkina, O. A. Neklyudova, V. F. Tishkin, “Application of the RKDG method for gas dynamics problems”, Math. Models Comput. Simul., 6:4 (2014), 397–407 | DOI | MR | MR

[7] Y. Cheng, C.-W. Shu, “Superconvergence and time evolution of discontinuous Galerkin finite element solutions”, J. Comput. Phys., 227 (2008), 9612–9627 | DOI | MR | Zbl

[8] X. Meng, C.-W. Shu, Q. Zhang, B. Wu, “Superconvergence of discontinuous Galerkin method for scalar nonlinear conservation laws in one space dimension”, SIAM J. Numer. Anal., 50 (2012), 2336–2356 | DOI | MR | Zbl

[9] B. van Leer, “Towards the ultimate conservative difference scheme. I: The quest for monotonicity”, Lect. Notes Phys., 18, 1973, 163–168 | DOI | Zbl

[10] B. van Leer, “Towards the ultimate conservative difference scheme. II: Monotonicity and conservation combined in a second-order scheme”, J. Comput. Phys., 14 (1974), 361–370 | DOI | MR | Zbl

[11] B. van Leer, “Towards the ultimate conservative difference scheme. III. Upstream-centered finitedifference schemes for ideal compressible flow”, J. Comput. Phys., 23 (1977), 263–275 | DOI | MR | Zbl

[12] B. van Leer, “Towards the ultimate conservative difference scheme. IV: A new approach to numerical convection”, J. Comput. Phys., 23 (1977), 276–299 | DOI | MR | Zbl

[13] B. van Leer, “Towards the ultimate conservative difference scheme. V: A second-order sequel to Godunov's method”, J. Comput. Phys., 32 (1979), 101–136 | DOI | MR

[14] V. P. Kolgan, “Application of the principle of minimizing the derivative to the construction of finite-difference schemes for computing discontinuous solutions of gas dynamics”, J. Comput. Phys., 230 (2011), 2384–2390 | DOI | MR | Zbl

[15] A. Harten, “High resolution schemes for hyperbolic conservation laws”, J. Comput. Phys., 49 (1983), 357–393 | DOI | MR | Zbl

[16] P. L. Roe, “Some contributions to the modelling of discontinuous flows”, Proceedings of the AMS/SIAM Seminar on Large Scale Computation in Fluid Mechanics (San Diego, 1983)

[17] R. F. Warming, R. W. Beam, “Upwind second order difference scheme with applications in aerodynamic flows”, AIAA Journal, 3 (1968), 176–189

[18] P. D. Lax, B. Wendroff, “Systems of conservation laws”, Pure Appl. Math., 13 (1960), 217–237 | DOI | MR | Zbl

[19] J. E. Fromm, “A method for reducing dispersion in convective difference schemes”, J. Comput. Phys., 3 (1968), 176–189 | DOI | Zbl

[20] C.-W. Shu, S. Osher, “Efficient implementation of essentially non-oscillatory shock-capturing schemes”, J. Comput. Phys., 77 (1988), 439–471 | DOI | MR | Zbl

[21] G. D. van Albada, B. van Leer, W. W. Roberts, “A comparative study of computational methods in cosmic gas dynamics”, Astron. Astrophys., 108 (1982), 76–84 | Zbl

[22] A. V. Rodionov, “Monotonic scheme of the second order of approximation for the continuous calculation of non-equilibrium flows”, USSR Comput. Math. Math. Phys., 27:2 (1987), 175–180 | DOI | MR | Zbl

[23] H. T. Huynh, “An upwind moment scheme for conservation laws”, Computational Fluid Dynamics 2004, Springer, Berlin, 2006, 761–766 | DOI

[24] Y. Suzuki, B. van Leer, An analysis of the upwind moment scheme and its extension to systems of nonlinear hyperbolic-relaxation equations, AIAA Paper No 2007-4468, 2007

[25] M. Ia. Ivanov, A. N. Kraiko, “The approximation of discontinuous solutions by using through calculation difference schemes”, USSR Comput. Math. Math. Phys., 18:3 (1978), 259–262 | DOI

[26] V. V. Ostapenko, “Convergence of finite-difference schemes behind a shock front”, Comput. Math. Math. Phys., 37:10 (1997), 1161–1172 | MR | Zbl

[27] O. A. Kovyrkina, V. V. Ostapenko, “On the practical accuracy of shock-capturing schemes”, Math. Models Comput. Simul., 6:2 (2014), 183–191 | DOI | MR | Zbl

[28] C.-W. Shu, S. Osher, “Efficient implementation of essentially non-oscillatory shock-capturing schemes, II”, J. Comput. Phys., 83 (1989), 32–78 | DOI | MR | Zbl

[29] H. T. Huynh, “Accurate upwind methods for the Euler equations”, SIAM J. Numer. Anal., 32:5 (1995), 1565–1619 | DOI | MR | Zbl

[30] A. Suresh, H. T. Huynh, “Accurate monotonicity-preserving schemes with Runge–Kutta time stepping”, J. Comput. Phys., 136 (1976), 83–99 | DOI | MR

[31] A. V. Rodionov, “A comparison of the CABARET and MUSCL-type schemes”, Math. Models Comput. Simul., 6:2 (2014), 203–225 | DOI | MR | MR

[32] P. R. Woodward, P. Colella, “The numerical simulation of two-dimensional fluid flow with strong shocks”, J. Comput. Phys., 54 (1984), 115–173 | DOI | MR | Zbl

[33] I. Yu. Tagirova, A. V. Rodionov, “Primenenie iskusstvennoi viazkosti v skhemakh tipa Godunova dlia borby s «karbunkul»-neustoichivostiu”, Matem. modelirovanie, 27:10 (2015), 47–64 | MR