Abstract
The classical Godunov scheme for the numerical solution of 3D gas dynamics equations is justified in the multidimensional case. An estimate is obtained for the error induced by replacing the exact solution of the multidimensional discontinuity breakup problem (known as the Riemann problem) with the solution of the 1D problems with the data on the left and right of the interface of each cell without considering the complicated flow in the neighborhood of the cells’ vertices. It is shown that, in the case of plane interfaces, the error has the first order of smallness in the time step and the approximate solution converges to the solution of semidiscrete equations as the time step vanishes. In fact, the time integration of these equations using the explicit Euler method represents the Godunov scheme.
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References
S. K. Godunov, “A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics,” Matem. Sb. 47 (89), 271–306 (1959).
O. M. Belotserkovskii, “On the calculation of the flow past axisymmetric bodies with detached shock waves using an electronic computer,” Prikl. Mat. Mekh. 24 (3), (1960).
Numerical Solution of Multidimensional Problems of Gas Dynamics, Ed. by S. K. Godunov (Nauka, Moscow, 1976) [in Russian].
S. K. Godunov, “Memories on difference schemes,” in Proceedings of the International Symposium on Godunov’s Method in Gas Dynamics, Michigan, 1997 (Nauch. Kniga, Novosibirsk, 1997), p. 40.
D. S. Balsara, “A two-dimensional HLLC Riemann solver for conservation laws: application to Euler and magnetohydrodynamic flows,” J. Comput. Phys. 231, 7476–7503 (2012).
D. S Balsara, M. Dumbser, and R. Abgrall, “Multidimensional HLLC Riemann solver for unstructured meshes-with application to Euler and MHD flows,” J. Comput. Phys. 261, 172–208 (2014).
L. Gosse, “A two-dimensional version of the Godunov scheme for scalar balance laws,” SIAM J. Numer. Anal. 52, 626–652 (2014).
M. E. Ladonkina and V. F. Tishkin, “On the connection of discontinuous Galerkin method and Godunov type methods of high order accuracy,” Preprint KIAM No. 49 (Keldysh Inst. Appl. Math., Moscow, 2014). http://library.keldysh.ru/preprint.asp?id=2014-49.
M. E. Ladonkina, O. A. Neklyudova, and V. F. Tishkin, “Application of the RKDG method for gas dynamics problems,” Math. Models Comput. Simul. 6, 397–407 (2014).
M. E. Ladonkina and V. F. Tishkin, “On Godunov-type methods of high order of accuracy,” Dokl. Math. 91, 189–192 (2015).
M. E. Ladonkina and V. F. Tishkin, “Godunov method: a generalization using piecewise polynomial approximations,” Differ. Equat. 51, 895–903 (2015).
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Original Russian Text © V.F. Tishkin, V.T. Zhukov, E.E. Myshetskaya, 2016, published in Matematicheskoe Modelirovanie, 2016, Vol. 28, No. 2, pp. 86–96.
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Tishkin, V.F., Zhukov, V.T. & Myshetskaya, E.E. Justification of Godunov’s scheme in the multidimensional case. Math Models Comput Simul 8, 548–556 (2016). https://doi.org/10.1134/S2070048216050124
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DOI: https://doi.org/10.1134/S2070048216050124