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Numerical Solution of the Baer–Nunziato Relaxation Model Using the Discontinuous Galerkin Method

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Abstract

We propose a numerical method for solving problems of two-phase two-velocity hydrodynamics in the framework of the Baer–Nunziato model with relaxation. The model equations are solved using the discontinuous Galerkin method with the WENO-S limiter, which is applied directly to the conservative variables of the model. Relaxation processes are modeled using the second-order implicit Runge–Kutta method with an adaptive choice of the integration step. The algorithm includes the Newton method for solving the equations of the nonlinear Runge–Kutta method. The corresponding Jacobi matrices are calculated using numerical differentiation. We present the results of numerical calculations that demonstrate the capabilities of our algorithm. The results of numerical calculations are compared with the analytical solution as well as with the results obtained by other authors. The results of numerical experiments with different relaxation rates are presented, including the case of “stiff” relaxation.

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Funding

The research by R.R. Tukhvatullina (the Introduction, Secs. 2.2 and 3, Appendix 1) was supported by the Russian Science Foundation, project no. 19-71-30004. The research by M.V. Alekseev and E.B. Savenkov (the Introduction, Secs. 1, 2.1, and 2.3, Appendix 2) was supported by the Moscow Center for Fundamental and Applied Mathematics, agreement with the RF Ministry of Science and Higher Education no. 075-15-2019-1623.

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  1. Keldysh Institute of Applied Mathematics, Moscow, 125047, Russia

    R. R. Tukhvatullina, M. V. Alekseev & E. B. Savenkov

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  1. R. R. Tukhvatullina
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  2. M. V. Alekseev
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  3. E. B. Savenkov
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Correspondence to R. R. Tukhvatullina, M. V. Alekseev or E. B. Savenkov.

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Translated by V. Potapchouck

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Tukhvatullina, R.R., Alekseev, M.V. & Savenkov, E.B. Numerical Solution of the Baer–Nunziato Relaxation Model Using the Discontinuous Galerkin Method. Diff Equat 57, 959–973 (2021). https://doi.org/10.1134/S0012266121070119

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  • DOI: https://doi.org/10.1134/S0012266121070119

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