Abstract
A method is proposed for constructing a CABARET scheme that approximates a hyperbolic system of conservation laws that cannot be written in the form of invariants. This technique is based on the method of quasi-invariants and additional flux correction, which ensures monotonization of the difference solution in calculating discontinuous solutions with shock waves and contact discontinuities. As an example, a system of conservation laws for nonisentropic gas dynamics with a polytropic equation of state is considered. Test calculations of the Blast Wave initial-boundary value problem showed that the proposed scheme suppresses nonphysical oscillations leading to the instability of the difference solution in the case when the CABARET scheme is used without additional flux correction.
Similar content being viewed by others
REFERENCES
B. L. Rozhdestvenskii and N. N. Yanenko, Systems of Quasilinear Equations and Their Applications to Gas Dynamics (Nauka, Moscow, 1978; Am. Math. Soc., Providence, 1983).
V. M. Goloviznin, M. A. Zaitsev, S. F. Karabasov, and I. A. Korotkin, New CFD Algorithms for Multiprocessor Computer Systems (Mosk. Gos. Univ., Moscow, 2013) [in Russian].
O. A. Kovyrkina and V. V. Ostapenko, “Monotonicity of the CABARET scheme approximating a hyperbolic equation with a sign-changing characteristic field,” Comput. Math. Math. Phys. 56 (5), 783–801 (2016).
N. A. Zyuzina and V. V. Ostapenko, “On the monotonicity of the CABARET scheme approximating a scalar conservation law with a convex flux,” Dokl. Math. 93 (1), 69–73 (2016).
V. V. Ostapenko and A. A. Cherevko, “Application of the CABARET scheme for calculation of discontinuous solutions of the scalar conservation law with nonconvex flux,” Dokl. Phys. 62 (10), 470–474 (2017).
O. A. Kovyrkina and V. V. Ostapenko, “Monotonicity of the CABARET scheme approximating a hyperbolic system of conservation laws,” Comput. Math. Math. Phys. 58 (9), 1435–1450 (2018).
T. S. Gologush, A. A. Cherevko, and V. V. Ostapenko, “Comparison of the WENO and CABARET schemes at calculation of the scalar conservation law with a nonconvex flux,” AIP Conf. Proc. 2293 (1), 370006 (2020).
G. S. Jiang and C. W. Shu, “Efficient implementation of weighted ENO schemes,” J. Comput. Phys. 126, 202–228 (1996).
P. Woodward and P. Colella, “The numerical simulation of two-dimensional fluid flow with strong shocks,” J. Comput. Phys. 54, 115–173 (1984).
Funding
This work was supported in part by the Russian Foundation for Basic Research and the National Natural Science Foundation of China, project no. 21-51-53012.
Author information
Authors and Affiliations
Corresponding authors
Ethics declarations
The authors declare that they have no conflicts of interest.
Additional information
Translated by I. Ruzanova
Rights and permissions
About this article
Cite this article
Ostapenko, V.V., Kolotilov, V.A. Application of the CABARET Scheme for Calculating Discontinuous Solutions of a Hyperbolic System of Conservation Laws. Dokl. Math. 104, 369–373 (2021). https://doi.org/10.1134/S1064562421060120
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1064562421060120