Verification of an entropy dissipative QGD-scheme for the 1D gas dynamics equations
Abstract
An entropy dissipative spatial discretization has recently been constructed for the multidimensional gas dynamics equations based on their preliminary parabolic quasi-gasdynamic (QGD) regularization. In this paper, an explicit finite-difference scheme with such a discretization is verified on several versions of the 1D Riemann problem, both well-known in the literature and new. The scheme is compared with the previously constructed QGD-schemes and its merits are noticed. Practical convergence rates in the mesh L1-norm are computed. We also analyze the practical relevance in the nonlinear statement as the Mach number grows of recently derived necessary conditions for L2-dissipativity of the Cauchy problem for a linearized QGD-scheme.
Keyword : 1D gas dynamics equations, entropy dissipative spatial discretization, explicit finite-difference scheme, verification on the Riemann problem, practical stability analysis
How to Cite
Zlotnik, A., & Lomonosov, T. (2019). Verification of an entropy dissipative QGD-scheme for the 1D gas dynamics equations. Mathematical Modelling and Analysis, 24(2), 179-194. https://doi.org/10.3846/mma.2019.013
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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A.A. Zlotnik. Quasi-gasdynamic system of equations with general equations of state. Dokl. Math., 81(2):312–316, 2010. https://doi.org/10.1134/s1064562410020419
A.A. Zlotnik. Spatial discretization of the one-dimensional quasi-gasdynamic system of equations and the entropy balance equation. Comput. Math. Math. Phys., 52(7):1060–1071, 2012. https://doi.org/10.1134/s0965542512070111
A.A. Zlotnik. Entropy-conservative spatial discretization of the multidimensional quasi-gasdynamic system of equations. Comput. Math. Math. Phys., 57(4):706–725, 2017. https://doi.org/10.1134/s0965542517020166
A.A. Zlotnik and B.N. Chetverushkin. Parabolicity of the quasi-gasdynamic system of equations, its hyperbolic second-order modification, and the stability of small perturbations for them. Comput. Math. Math. Phys., 48(3):420–446, 2008. https://doi.org/10.1134/s0965542508030081
A.A. Zlotnik and T.A. Lomonosov. On conditions for L2-dissipativity of linearized explicit QGD–finite-difference schemes for the equations of one-dimensional gas dynamics. Dokl. Math., 98(2):458–463, 2018. https://doi.org/10.1134/S1064562418060200
A.A. Zlotnik and T.A. Lomonosov. Conditions for L2-dissipativity of linearized explicit finite-difference schemes with regularization for the equations of 1D barotropic gas dynamics. Comput. Math. Math. Phys., 59, 2019. (accepted).
B.N. Chetverushkin. Kinetic schemes and quasi-gas dynamic system of equations.CIMNE, Barcelona, 2008.
T.G. Elizarova. Quasi-gas dynamic equations. Springer, Dordrecht, 2009. https://doi.org/10.1007/978-3-642-00292-2
T.G. Elizarova and E.V. Shil’nikov. Capabilities of a quasi-gasdynamic algorithm as applied to inviscid gas flow simulation. Comput. Math. Math. Phys.,49(3):532–548, 2009.https://doi.org/10.1134/s0965542509030142
T.G. Elizarova and I.A. Shirokov. Regularized equations and examples of their using in modeling gas-dynamic flows. MAKS Press, Moscow, 2017. (in Russian)
U.S. Fjordholm, S. Mishra and E. Tadmor. Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws. SIAM J. Numer. Anal., 50:544–573, 2012. https://doi.org/10.1137/110836961
V.A. Gavrilin and A.A Zlotnik. On spatial discretization of the one-dimensional quasi-gasdynamic system of equations with general equations of state and entropy balance. Comput. Math. Math. Phys., 55(2):264–281, 2015. https://doi.org/10.1134/s0965542515020098
S.K. Godunov and I.M. Kulikov. Computation of discontinuous solutions of fluid dynamics equations with entropy nondecrease guarantee. Comput. Math. Math. Phys., 54(6):1012–1024, 2014. https://doi.org/10.1134/s0965542514060086
M.V. Kraposhin, E.V. Smirnova, T.G. Elizarova and M.A. Istomina. Development of a new OpenFOAM solver using regularized gas dynamic equations. Comput. Fluids, 166:163–175, 2018. https://doi.org/10.1016/j.compfluid.2018.02.010
A.G. Kulikovskii, N.V. Pogorelov and A.Yu. Semenov. Mathematical aspects of numerical solution of hyperbolic systems. Chapman&Hall/CRC, London, 2001.
R.J. LeVeque. Finite volume methods for hyperbolic problems. Cambridge University Press, Cambridge, 2004. https://doi.org/10.1017/cbo9780511791253.005
R. Liska and B. Wendroff. Comparison of several difference schemes on 1D and 2D test problems for the Euler equations. SIAM J. Sci. Comput., 25(3):995–1017, 2003. https://doi.org/10.1007/978-3-642-55711-878
G.P. Prokopov. Necessity of entropy control in gasdynamic computations. Comput. Math. Math. Phys., 47(9):1528–1537, 2007. https://doi.org/10.1134/s0965542507090138
M. Svärd and H.Özcan. Entropy stable schemes for the Euler equations with far-field and wall boundary conditions. J. Sci. Comput., 58(1):61–89, 2014.https://doi.org/10.1007/s10915-013-9727-7
E. Tadmor. Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numer., 12:451–512, 2003. https://doi.org/10.1017/cbo9780511550157.007
E.F. Toro. Riemann solvers and numerical methods for fluid dynamics, 3rd ed. Springer, Berlin, 2009. https://doi.org/10.1007/978-3-662-03915-1
A.R. Winters and G.J. Gassner. Affordable, entropy conserving and entropy stable flux functions for the ideal MHD equations. J. Comput. Phys., 304:72–108, 2016. https://doi.org/10.1016/j.jcp.2015.09.055
A.A. Zlotnik. Quasi-gasdynamic system of equations with general equations of state. Dokl. Math., 81(2):312–316, 2010. https://doi.org/10.1134/s1064562410020419
A.A. Zlotnik. Spatial discretization of the one-dimensional quasi-gasdynamic system of equations and the entropy balance equation. Comput. Math. Math. Phys., 52(7):1060–1071, 2012. https://doi.org/10.1134/s0965542512070111
A.A. Zlotnik. Entropy-conservative spatial discretization of the multidimensional quasi-gasdynamic system of equations. Comput. Math. Math. Phys., 57(4):706–725, 2017. https://doi.org/10.1134/s0965542517020166
A.A. Zlotnik and B.N. Chetverushkin. Parabolicity of the quasi-gasdynamic system of equations, its hyperbolic second-order modification, and the stability of small perturbations for them. Comput. Math. Math. Phys., 48(3):420–446, 2008. https://doi.org/10.1134/s0965542508030081
A.A. Zlotnik and T.A. Lomonosov. On conditions for L2-dissipativity of linearized explicit QGD–finite-difference schemes for the equations of one-dimensional gas dynamics. Dokl. Math., 98(2):458–463, 2018. https://doi.org/10.1134/S1064562418060200
A.A. Zlotnik and T.A. Lomonosov. Conditions for L2-dissipativity of linearized explicit finite-difference schemes with regularization for the equations of 1D barotropic gas dynamics. Comput. Math. Math. Phys., 59, 2019. (accepted).