Hybrid Characteristic Method based on the Discontinuous Galerkin Method for Solving Gas Dynamics Problems

Abstract

The article presents a numerical method for solving the compressible perfect inviscid gas dynamics problems in one-dimensional and two-dimensional spatial formulations. The method uses the discontinuous Galerkin method approach to approximate the solution over space and the solution transfer along the characteristics technique to integrate the equations over time. This approach allows one to get rid of the standard multi-stage procedure of the Runge --- Kutta method and significantly reduce the computational complexity of the method. Instead of integration over time integration of flux function over the spatial domain of dependence on the previous time layer was applied. A generalization of the method to the two-dimensional case for an arbitrary grid cell shape is presented. Test calculations were carried out on a number of one-dimensional and two-dimensional problems with both smooth and discontinuous solutions. Tests have shown that the method has second-order accuracy when using piecewise linear reconstruction of the solution on a cell and third-order accuracy when using quadratic polynomials. In addition, the method has high resolution and low dissipation. The method allows one to resolve gas instabilities such as the Richtmyer --- Meshkov instability. Also, despite the high order of accuracy the method does not require the use of special procedures for monotonizing the solution Discontinuous

Please cite this article in English as:

Lukin V.V. Hybrid characteristic method based on the discontinuous Galerkin method for solving gas dynamics problems. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2025, no. 1 (118), pp. 46--64 (in Russ.). EDN: EGAMAE

References

[1] Zarubin V.S., Kuvyrkin G.N. Matematicheskie modeli mekhaniki i elektrodinamiki sploshnoy sredy [Mathematical models of continuum mechanics and electrodynamics]. Moscow, BMSTU Publ., 2008.

[2] Lukin V., Postnov K., Shakura N. Modeling of the L1 point jet formation in binary star Her X1. Lobachevskii J. Math., 2024, vol. 45, no. 1, pp. 85--94. DOI: https://doi.org/10.1134/S1995080224010360

[3] Galanin M.P., Lukin V.V., Chechetkin V.M. 3D hydrodynamical simulation of accretion disk in binary star system using RKDG CFD solver. J. Phys.: Conf. Ser., 2018, vol. 1103, art. 012019. DOI: http://dx.doi.org/10.1088/1742-6596/1103/1/012019

[4] Shu C.-W., Qiu J. A comparison of troubled-cell indicators for Runge --- Kutta discontinuous Galerkin methods using weighted essentially nonoscillatory limiters. SIAM J. Sci. Comput., 2005, vol. 27, iss. 3, pp. 995--1013. DOI: https://doi.org/10.1137/04061372X

[5] Qiu J., Shu C.-W. Runge --- Kutta discontinuous Galerkin method using WENO limiters. SIAM J. Sci. Comput., 2005, vol. 26, iss. 3, pp. 907--929. DOI: https://doi.org/10.1137/S1064827503425298

[6] Qiu J.-X., Shu C.-W. Hermite WENO schemes and their application as limiters for Runge --- Kutta discontinuous Galerkin method: one-dimensional case. J. Comput. Phys., 2003, vol. 193, iss. 1, pp. 115--135. DOI: https://doi.org/10.1016/j.jcp.2003.07.026

[7] Jiang G., Shu C.-W. Efficient implementation of weighted ENO schemes. J. Comput. Phys., 1996, vol. 126, iss. 1, pp. 202--228. DOI: https://doi.org/10.1006/jcph.1996.0130

[8] Galepova V.D., Lukin V.V., Marchevskiy I.K., et al. Comparative study of WENO and Hermite WENO limiters for gas flows numeriсal simulations using the RKDG method. Preprinty IPM im. M.V. Keldysha [KIAM Preprint], 2017, no. 131 (in Russ.). DOI: http://doi.org/10.20948/prepr-2017-131

[9] Lukin V.V., Korchagova V.N., Sautkina S.M. On stable Runge --- Kutta methods for solving hyperbolic equations by the discontinuous Galerkin method. Diff. Equat., 2021, vol. 57, no. 7, pp. 921--933. DOI: https://doi.org/10.1134/S0012266121070089

[10] Lukin V.V. Stable Runge --- Kutta methods of 2nd and 3rd order for gas dynamics simulation using discontinuous Galerkin method. Preprinty IPM im. M.V. Keldysha [KIAM Preprint], 2022, no. 52 (in Russ.). DOI: https://doi.org/10.20948/prepr-2022-52

[11] Gatsuk V.K., Lukin V.V. Mixed characteristic discontinuous Galerkin approach for perfect gas dynamics modeling. J. Phys.: Conf. Ser., 2021, vol. 2028, art. 012018. DOI: http://dx.doi.org/10.1088/1742-6596/2028/1/012018

[12] Popov Yu.P., Samarskii A.A. Completely conservative difference schemes for the equations of gas dynamics in Euler’s variables. U.S.S.R. Comput. Math. Math. Phys., 1970, vol. 10, iss. 3, pp. 265--273. DOI: https://doi.org/10.1016/0041-5553(70)90133-3

[13] Toro E.F. Riemann solvers and numerical methods for fluid dynamics. Berlin, Heidelberg, Springer, 2013. DOI: https://doi.org/10.1007/b79761

[14] Kulikovskiy A.G., Pogorelov N.V., Semenov A.Yu. Matematicheskie voprosy chislennogo resheniya giperbolicheskikh sistem uravneniy [Mathematical questions of hyperbolic equations systems numerical solution]. Moscow, FIZMATLIT Publ., 2001.

[15] Popov M.V., Ustyugov S.D. Piecewise parabolic method on local stencil for gasdynamic simulations. Comput. Math. and Math. Phys., 2007, vol. 47, no. 12, pp. 1970--1989. DOI: https://doi.org/10.1134/S0965542507120081

[16] Collela P., Woodward P. The Piecewise Parabolic Method (PPM) for gas-dynamical simulations. J. Comp. Phys., 1984, vol. 54, iss. 1, pp. 174--201. DOI: https://doi.org/10.1016/0021-9991(84)90143-8

[17] Ustyugov C.D., Popov M.B. Piecewise parabolic method on local stencil. I. Advection equation and Burgers’ equation. Preprinty IPM im. M.V. Keldysha [KIAM Preprint], 2006, no. 65 (in Russ.).

[18] Uctyugov C.D., Popov M.B. Piecewise parabolic method on local stencil. II. Gasdynamics equations. Preprinty IPM im. M.V. Keldysha [KIAM Preprint], 2006, no. 71 (in Russ.).

[19] Kurganov A., Tadmor E. Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers. Numer. Methods Partial Differ. Equ., 2002, vol. 18, iss. 5, pp. 584--608. DOI: https://doi.org/10.1002/num.10025

[20] Helzel C., Kerkmann D., Scandurra L. A new ADER method inspired by the active flux method. J. Sci. Comput., 2019, vol. 80, no. 3, pp. 1463--1497. DOI: https://doi.org/10.1007/s10915-019-00988-1

[21] Godunov S.K. A finite difference method for the computation of discontinuous solutions of the equations of fluid dynamics. Sbornik: Mathematics, 1959, vol. 47, no. 8-9, pp. 357--393.