Geodesic
Hybrid numerical flux for solving the problems of supersonic flow of solid bodies
Matematičeskoe modelirovanie, Tome 33 (2021) no. 5, pp. 47-56.

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

Numerical modeling of supersonic flow around solid bodies is a complex task due to the possibility of nonphysical instabilities that can affect the solution. In this paper, we propose a new hybrid numerical flux for calculating the fluxes of the Euler part of the Navier–Stokes system of equations. This flux avoids the occurrence of instability and maintains high accuracy on shock waves and boundary layers. This flux is a combination of Godunov's numerical flow and Rusanov–Lax–Friedrichs numerical flow. Numerical simulation of supersonic flow around a cruise missile Tomahawk has been carried out.
Keywords: hypersonic gas dynamics, numerical flux, discontinuous Galerkin method. 
@article{MM_2021_33_5_a3,
     author = {M. E. Ladonkina and O. A. Nekliudova and V. F. Tishkin},
     title = {Hybrid numerical flux for solving the problems of supersonic flow of solid bodies},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {47--56},
     publisher = {mathdoc},
     volume = {33},
     number = {5},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2021_33_5_a3/}
}
TY  - JOUR
AU  - M. E. Ladonkina
AU  - O. A. Nekliudova
AU  - V. F. Tishkin
TI  - Hybrid numerical flux for solving the problems of supersonic flow of solid bodies
JO  - Matematičeskoe modelirovanie
PY  - 2021
SP  - 47
EP  - 56
VL  - 33
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MM_2021_33_5_a3/
LA  - ru
ID  - MM_2021_33_5_a3
ER  - 
%0 Journal Article
%A M. E. Ladonkina
%A O. A. Nekliudova
%A V. F. Tishkin
%T Hybrid numerical flux for solving the problems of supersonic flow of solid bodies
%J Matematičeskoe modelirovanie
%D 2021
%P 47-56
%V 33
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MM_2021_33_5_a3/
%G ru
%F MM_2021_33_5_a3
M. E. Ladonkina; O. A. Nekliudova; V. F. Tishkin. Hybrid numerical flux for solving the problems of supersonic flow of solid bodies. Matematičeskoe modelirovanie, Tome 33 (2021) no. 5, pp. 47-56. http://geodesic.mathdoc.fr/item/MM_2021_33_5_a3/

[1] A. V. Rodionov, “Artificial viscosity to cure the shock instability in high-order Godunov-type schemes”, Computers and Fluids, 190 (2019), 77–97 | DOI | MR | Zbl

[2] M. Pandolfi, D. D'Ambrosio, “Numerical Instabilities in Upwind Methods: Analysis and Cures for the “Carbuncle” Phenomenon”, J. of Computational Physics, 166 (2001), 271–301 | DOI | MR | Zbl

[3] S. Osher, F. Solomon, “Upwind Difference Schemes for Hyperbolic Systems of Conservation Laws”, Mathematics of computation, 38 (1982), 339–374 | DOI | MR | Zbl

[4] M. Pandolfi, “A contribution to the numerical prediction of unsteady flows”, AIAA journal, 22 (1984), 602–610 | DOI | Zbl

[5] P. L. Roe, “Approximate Riemann solvers, parameter vectors, and difference schemes”, Journal of Computational Physics, 43 (1981), 357–372 | DOI | MR | Zbl

[6] S. K. Godunov, “A finite difference method for the computation of discontinuous solutions of the equations of fluid dynamics”, Sbornik: Mathematics, 47:8-9 (1959), 357–393 | MR | Zbl

[7] E. F. Toro, Riemann solvers and numerical methods for fluid dynamics, Third Edition, Springer, 2010 | MR

[8] V. V. Rusanov, “The calculation of the interaction of non-stationary shock waves and obstacles”, USSR Comp. Mathematics and Mathematical Physics, 1:2 (1962), 304–320 | DOI | MR

[9] P. D. Lax, “Weak solutions of nonlinear hyperbolic equations and their numerical computation”, Communications on Pure and Applied Mathematics, 7:1 (1954), 159–193 | DOI | MR | Zbl

[10] B. Van Leer, Upwind and High-Resolution Schemes, Springer, 1997, 80–89

[11] M.-S. Liou, “A Sequel to AUSM: AUSM$^+$”, J. of Comp. Physics, 129 (1996), 364–382 | DOI | MR | Zbl

[12] H. Nishikawa, K. Kitamura, “Very simple, carbuncle-free, boundary-layer-resolving, rotated-hybrid Riemann solvers”, J. of Computational Physics, 227 (2008), 2560–2581 | DOI | MR | Zbl

[13] S. Guo, W. Q. Tao, “A hybrid flux splitting method for compressible flow”, Numerical Heat Transfer, Part B: Fundamentals, 73 (2018), 33–47 | DOI

[14] L. J. Hu, L. Yuan, “A robust hybrid HLLC-force scheme for curing numerical shock instability”, Applied Mechanics and Materials, 577 (2014), 749–753 | DOI

[15] A. Ferrero, D. D'Ambrosio, “An Hybrid Numerical Flux for Supersonic Flows with Ap-plication to Rocket Nozzles”, 17$^{\text{TH}}$ International Conference of Numerical Analysis and Applied Mathematics (Sept. 2019, Rhodes, Greece), 23–28

[16] P. Woodward, Ph. Colella, “The numerical simulation of two-dimensional fluid flow with strong shocks”, J. of Comp. Phys., 54:1 (1984), 115–173 | DOI | MR | Zbl

[17] B. Cockburn, “An Introduction to the Discontinuous Galerkin Method for Convection-Dominated Problems”, Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics, 1697, 1998, 151–268 | MR | Zbl

[18] F. Bassi, S. Rebay, “Numerical evaluation of two discontinuous Galerkin methods for the compressible Navier-Stokes equations”, Int. J. Numer. Meth. Fluids, 40 (2002), 197–207 | DOI | MR | Zbl

[19] K. Yasue, M. Furudate, N. Ohnishi, K. Sawada, “Implicit Discontinuous Galerkin Method for RANS Simulation Utilizing Pointwise Relaxation Algorithm”, Comm. Comp. Phys., 7:3 (2010), 510–533 | DOI | MR | Zbl

[20] R. J. Spiteri, S. J. Ruuth, “A New Class of Optimal High-Order Strong Stability-Preserving Time Discretization Methods”, SIAM J. Numer. Anal., 40:2 (2002), 469–491 | DOI | MR | Zbl

[21] M. M. Krasnov, P. A. Kuchugov, M. E. Ladonkina, V. F. Tishkin, “Discontinuous Galerkin method on three-dimensional tetrahedral meshes. The usage of the operator programming method”, Math. Models Comput. Simul., 9:5 (2017), 529–543 | DOI | MR | Zbl

[22] M. E. Ladonkina, O. A. Nekliudova, V. V. Ostapenko, V. F. Tishkin, “On the accuracy of the discontinuous Galerkin method in the calculation of shock waves”, JCM, 58:8 (2018), 148–156 | MR