Application of the principle of minimizing the derivative to the construction of finite-difference schemes for computing discontinuous solutions of gas dynamics. (Reprint).

*(English. Russian original)*Zbl 1316.76063
J. Comput. Phys. 230, No. 7, 2384-2390 (2011); translation from Uchenie Zapiski TsAGI 1972, No. 3(6), 68-76 (1972).

Summary: Progress in computer technology now allows the application of numerical methods to some practical multidimensional steady and unsteady gas flows. While selecting a numerical method, preference should be given to higher-order schemes, since they allow the study of the fine features of the solution and the reduction of the computation time. In general, however, solutions of non-stationary nonlinear equations of gas dynamics are not smooth and may include strong discontinuities, such as shock waves or contact discontinuities. This fact is important for choosing an appropriate numerical scheme. If the strong discontinuities are extracted from the simulation domain and traced (shock fitting), while the rest of the flow is smooth, it is possible to use numerical schemes with approximation close to the second order. However, if the number of discontinuities increases with time leading to a complicated pattern, the application of the shock-fitting approach becomes difficult. In this case it is preferable to apply shock-capturing schemes, which have a lower order of approximation but allow the integration of the governing equations through the discontinuities without extracting their surfaces.
Shock-capturing schemes of a higher order of approximation are currently under development.

In this paper, certain aspects of constructing a numerical scheme applicable to discontinuous flows of gas dynamics are discussed using the idea of minimizing the derivatives of the solution. A numerical scheme that is second-order accurate in space and first-order accurate in time is developed for the model equation \(u_t + u_x = 0\). The properties of this scheme are analyzed and the conditions for its stability and monotonicity are formulated. These results are then used in devising a similar scheme for the non-stationary one-dimensional equations of gas dynamics. The performance of the new scheme is compared to that of other schemes, using the break-up of an initial discontinuity (Riemann problem) as an example.

In this paper, certain aspects of constructing a numerical scheme applicable to discontinuous flows of gas dynamics are discussed using the idea of minimizing the derivatives of the solution. A numerical scheme that is second-order accurate in space and first-order accurate in time is developed for the model equation \(u_t + u_x = 0\). The properties of this scheme are analyzed and the conditions for its stability and monotonicity are formulated. These results are then used in devising a similar scheme for the non-stationary one-dimensional equations of gas dynamics. The performance of the new scheme is compared to that of other schemes, using the break-up of an initial discontinuity (Riemann problem) as an example.

##### MSC:

76M20 | Finite difference methods applied to problems in fluid mechanics |

76N15 | Gas dynamics (general theory) |

PDF
BibTeX
XML
Cite

\textit{V. P. Kolgan}, J. Comput. Phys. 230, No. 7, 2384--2390 (2011; Zbl 1316.76063); translation from Uchenie Zapiski TsAGI 1972, No. 3(6), 68--76 (1972)

Full Text:
DOI

##### References:

[1] | Lax, P.D.; Wendroff, B., Systems of conservation laws, Commun. pure appl. math., 13, 2, 217, (1960) · Zbl 0152.44802 |

[2] | Babenko, K.I.; Voskresenkii, G.P., Numerical method for computing three-dimensional supersonic flow past bodies, Zh. vychsl. mat. mat. fiz., 1, 6, 1051, (1961), [in Russian]; translation, USSR Comput. Math. Math. Phy., 1 (6), 1220, 1961 |

[3] | Alalikin, G.B.; Godunov, S.K.; Kireeva, I.L.; Pliner, L.A., Solution of one-dimensional gas-dynamic problems using moving grids, (1970), Nauka Moscow, [in Russian] |

[4] | Lax, P.D., Weak solutions of nonlinear hyperbolic equations and their numerical computations, Commun. pure appl. math., 7, 1, 159, (1954) · Zbl 0055.19404 |

[5] | Godunov, S.K., A finite-difference method for the numerical computation of discontinuous solutions of the equations of gas-dynamics, Math. sb., 47, 271, (1959), [in Russian]; translation, US Joint Publ. Res. Service, JPRS 7226, 1969 · Zbl 0171.46204 |

[6] | Belotserkovskii, O.M.; Davidov, Yu. M., Non-stationary method of “large particlesâ€ť for gas-dynamics problems, Zh. vychisl. mat. mat. fiz., 11, 1, 182, (1971), [in Russian]; translation, USSR Comput. Math. Math. Phys., 11 (1), 241, 1961 |

[7] | Rusanov, V.V., Third order difference schemes for direct computing of discontinuous solutions, Dokl. akad. nauk SSSR, 180, 6, 1303, (1968), [in Russian]; translation, Soviet Math. Dokl., 9 (3), 771, 1968 |

[8] | Balakin, V.B., On runge – kutta-type methods for gas dynamics, Zh. vychisl. mat. mat. fiz., 10, 6, 1512, (1971), [in Russian]; translation, USSR Comput. Math. Math. Phys., 10 (6), 208, 1961 · Zbl 0205.56504 |

[9] | Kosarev, V.I., Computation of supersonic steady flows with internal shock waves, Zh. vychisl. mat. mat. fiz., 11, 5, 1262, (1971), [in Russian]; translation, USSR Comput. Math. Math. Phys., 11 (5), 208, 1971 |

[10] | Harlow, F.H., Hydrodynamic problems involving large fluid distortions, J. assoc. comput. Mach., 4, 2, 137, (1957) |

[11] | Godunov, S.K.; Zabrodin, A.V.; Prokopov, G.P., A difference scheme for two-dimensional problems of gas dynamics and a computation of a flow with a detached shock, Zh. vychisl. mat. mat. fiz., 1, 6, 1020, (1961), [in Russian]; translation, USSR Comput. Math. Math. Phys., 1 (6), 1187, 1961 · Zbl 0146.23004 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.