Godunov method: a generalization using piecewise polynomial approximations. (English. Russian original) Zbl 1329.76180
Differ. Equ. 51, No. 7, 895-903 (2015); translation from Differ. Uravn. 51, No. 7, 899-907 (2015).
Summary: We show that the schemes of the Galerkin discontinuous method can be treated as a generalization of the Godunov method to piecewise polynomial functions. Using the problem on the interaction of a shock wave with an entropy perturbation as an example, we obtain nearly second order accuracy in the wake of the shock wave.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 76N15 | Gas dynamics (general theory) |
| 76L05 | Shock waves and blast waves in fluid mechanics |
References:
| [1] | Godunov, S.K., A Difference Method for Numerical Calculation of Discontinuous Solutions of the Equations of Hydrodynamics. Mat. Sb., 1959, vol. 47, no. 3, pp. 271-306. · Zbl 0171.46204 |
| [2] | Godunov, S.K., Zabrodin, A.V., and Ivanov, M.Ya., et al., Chislennoe reshenie mnogomernykh zadach gazovoi dinamiki (Numerical Solution of Multidimensional Problems of Gas Dynamics), Moscow Nauka, 1976. |
| [3] | Courant, R., Partial Differential Equations (Courant, R. and Hilbert, D., Methods of Mathematical Physics, vol. 2), New York, 1962. Translated under the title Uravneniya s chastnymi proizvodnymi, Moscow Mir, 1964. · Zbl 0099.29504 |
| [4] | Men’shov, I.S., Increase in the Approximation Order for Godunov’s Scheme on the Basis of the Solution of the Generalized Riemann Problem, Zh. Vychisl. Mat. Mat. Fiz., 1990, vol. 30, no. 9, pp. 1357-1371. · Zbl 0714.65080 |
| [5] | Cockburn, B., An Introduction to the Discontinuous Galerkin Method for Convection-Dominated Problems, Advanced Numerical Approximation of Nonlinear Hyperbolic Equations: Lecture Notes in Mathematics, 1998, vol. 1697, pp. 151-268. · Zbl 0927.65120 · doi:10.1007/BFb0096353 |
| [6] | Ladonkina, M.E., Neklyudova, O.A., and Tishkin, V.F., Investigation of Galerkin Discontinuous Method for the Solution of Problems of Gas Dynamics, Mat. Model., 2014, vol. 26, no. 1, pp. 17-32. · Zbl 1313.76090 |
| [7] | Ladonkina, M.E. and Tishkin, V.F., On the Relationship of the Galerkin Method and Godunov-Type Methods of Higher-Order Accuracy, Preprint Inst. Appl. Math., Moscow, 2014, no. 49. · Zbl 1319.76028 |
| [8] | Osher, S. and Chakravarthy, S.R., High Resolution Schemes and the Entropy Condition, SIAM J. Numer. Anal., 1984, vol. 21, pp. 955-984. · Zbl 0556.65074 · doi:10.1137/0721060 |
| [9] | Vyaznikov, K.V., Tishkin, V.F., and Favorskii, A.P., Construction of Monotone Difference Schemes of Increased Order of Approximation for Systems of Equations of Hyperbolic Type, Mat. Model., 1989, vol. 1, no. 5, pp. 95-120. · Zbl 0972.76521 |
| [10] | Ladonkina, M.E., Neklyudova, O.A., and Tishkin, V.F., Investigation of the Influence of a Limiter on the Accuracy Order of the Solution by the Galerkin Discontinuous Method, Mat. Model., 2012, vol. 24, no. 12, pp. 124-128. · Zbl 1313.76089 |
| [11] | Ladonkina, M.E., Neklyudova, O.A., and Tishkin, V.F., Investigation of the Influence of a Limiter on the Accuracy Order of the Solution by the Galerkin Discontinuous Method, Preprint Inst. Appl. Math., Moscow, 2012, no. 31. · Zbl 1313.76089 |
| [12] | Ladonkina, M.E., Neklyudova, O.A., and Tishkin, V.F., Limiter of Higher-Order Accuracy for the Galerkin Discontinuous Method on Triangular Grids, Preprint Inst. Appl. Math., Moscow, 2013, no. 53. · Zbl 1313.76090 |
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