Abstract
For problems with complex geometry, a numerical method is proposed for solving the three-dimensional nonstationary Euler equations on Cartesian grids with the use of hybrid computing systems. The baseline numerical scheme, a method for implementing internal boundary conditions on body-unfitted grids, and an iterative matrix-free LU-SGS method for solving the discretized equations are described. An efficient software implementation of the numerical algorithm on a multiprocessor hybrid CPU/GPU computing system is considered. Results of test computations are presented.
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Lin Fu, Zhenghong Gao, Kan Xu, and Fang Xu, “A multi-block viscous flow solver based on GPU parallel methodology,” Comput. Fluids 95, 19–39 (2014).
A. Jameson and E. Turkel, “Implicit schemes and LU decomposition,” Math. Comput. 37 (156), 385–397 (1981).
V. P. Kolgan, “The principle of minimal derivative values as applied to the construction of finite-difference schemes for the computation of discontinuous gas flows,” Uch. Zap. Tsentr. Aerogidrodin. Inst. 3 (6), 68–77 (1972).
W. K. Anderson, J. L. Thomas, and B. van Leer, “Comparison of finite volume flux vector splitting for the Euler equations,” AIAA J. 24 (9), 1453–1460 (1986).
B. van Leer, “Towards the ultimate conservative difference scheme: V. A second-order sequel to Godunov’s method,” J. Comput. Phys. 32, 101–136 (1979).
J. E. Fromm, “A method for reducing dispersion in convective difference schemes,” J. Comput. Phys. 3, 176–187 (1968).
S. K. Godunov and V. S. Ryaben’kii, Difference Schemes (Nauka, Moscow, 1977) [in Russian].
G. D. van Albada, B. van Leer, and W. Roberts, “A comparative study of computational methods in cosmic gas dynamics,” Astron. Astrophys. 108, 76–84 (1982).
S. K. Godunov, “Difference method for computing discontinuous solutions of fluid dynamics equations,” Mat. Sb. 47 (3), 271–306 (1959).
S. K. Godunov, A. V. Zabrodin, M. Ya. Ivanov, A. N. Kraiko, and G. P. Prokopov, Numerical Solution of Multidimensional Problems in Gas Dynamics (Nauka, Moscow, 1976) [in Russian].
E. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics (Springer, Berlin, 2009).
V. V. Rusanov, “Third-order accurate shock-capturing schemes for computing discontinuous solutions,” Dokl. Akad. Nauk SSSR 180, 1303–1305 (1968).
I. Menshov and Y. Nakamura, “Hybrid explicit-implicit, unconditionally stable scheme for unsteady compressible flows,” AIAA J. 42 (3), 551–559 (2004).
I. S. Menshov and M. A. Kornev, “Free-boundary method for the numerical solution of gas-dynamic equations in domains with varying geometry,” Math. Model. Computer Simul. 6 (6), 612–621 (2014).
I. S. Menshov and P. V. Pavlukhin, Preprint No. 92, IPM RAN (Keldysh Inst. of Applied Mathematics, Russian Academy of Sciences, Moscow, 2014).
O. Boiron, G. Chiavassa, and R. Donat, “A high-resolution penalization method for large Mach number flows in the presence of obstacles,” Comput. Fluids 38, 703–714 (2009).
L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Nauka, Moscow, 1986; Butterworth-Heinemann, Oxford, 1987).
I. Menshov and Y. Nakamura, “An implicit advection upwind splitting scheme for hypersonic air flows in thermochemical nonequilibrium,” Collection of Technical Papers of 6th International Symposium on CFD (Lake Tahoe, Nevada 1995), pp. 815–821.
P. V. Pavlukhin, “Implementation of a parallel LU-SGS method for gas dynamic applications on cluster GPU systems,” Vestn. Nizhegorod. Gos. Univ., No. 1, 213–218 (2013).
A. Jameson, “Airfoil admitting nonunique solutions to the Euler equations,” AIAA Paper, No. 91-1625 (1991).
M. M. Hafez and W. H. Guo, “Nonuniqueness of transonic flows,” Acta Mech. 138 (3), 177–184 (1999).
A. G. Kuzmin, “Instability and bifurcation of transonic flow over airfoils,” AIAA Paper (2004).
K. R. Laflin, S. M. Klausmeyer, T. Zickuhr, et al., “Data summary from second AIAA computational fluid dynamics drag prediction workshop,” J. Aircraft 42 (5), 1165–1178 (2005).
V. E. Borisov, A. A. Davydov, I. Yu. Kudryashov, A. E. Lutskii, and I. S. Menshov, “Parallel implementation of an implicit scheme based on the LU-SGS method for 3D turbulent flows,” Math. Model. Computer Simul. 7 (3), 222–232 (2015).
A. E. Lutskii and A. V. Severin, Preprint No. 38, IPM RAN (Keldysh Inst. of Applied Mathematics, Russian Academy of Sciences, Moscow, 2013).
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Original Russian Text © I.S. Menshov, P.V. Pavlukhin, 2016, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2016, Vol. 56, No. 9, pp. 1677–1691.
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Menshov, I.S., Pavlukhin, P.V. Efficient parallel shock-capturing method for aerodynamics simulations on body-unfitted cartesian grids. Comput. Math. and Math. Phys. 56, 1651–1664 (2016). https://doi.org/10.1134/S096554251609013X
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DOI: https://doi.org/10.1134/S096554251609013X