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On \(L^2\)-dissipativity of linearized explicit finite-difference schemes with a regularization on a non-uniform spatial mesh for the 1D gas dynamics equations. (English) Zbl 1411.65118
Summary: We deal with an explicit finite-difference scheme with a regularization for the 1D gas dynamics equations linearized at the constant solution. The sufficient condition on the Courant number for the \(L^2\)-dissipativity of the scheme is derived in the case of the Cauchy problem and a non-uniform spatial mesh. The energy-type technique is developed to this end, and the proof is both short and under clear conditions on matrices of the convective and regularizing (dissipative) terms. A scheme with a kinetically motivated regularization is considered as an application in more detail.

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
76M20 Finite difference methods applied to problems in fluid mechanics
76N15 Gas dynamics, general
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
Full Text: DOI
[1] Richtmyer, R. D.; Morton, K. W., Difference Methods for Initial-Value Problems, (1967), Wiley-Interscience · Zbl 0155.47502
[2] Godunov, S. K.; Ryabenkii, V. S., Difference schemes, (Studies in Mathematics and its Applications, vol. 19, (1987), North Holland: North Holland Amsterdam)
[3] LeVeque, R. J., Finite Volume Methods for Hyperbolic Problems, (2004), Cambridge University Press: Cambridge University Press Cambridge
[4] (Abgrall, R.; Shu, C.-W., Handbook of Numerical Methods for Hyperbolic Problems. Handbook of Numerical Methods for Hyperbolic Problems, Handbook of Numerical Analysis, vol. 17, (2016), North Holland: North Holland Amsterdam) · Zbl 1352.65001
[5] Zlotnik, A. A., 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) · Zbl 1274.35301
[6] Chetverushkin, B. N., Kinetic Schemes and Quasi-Gasdynamic System of Equations, (2008), CIMNE: CIMNE Barcelona
[7] Elizarova, T. G., Quasi-Gas Dynamic Equations, (2009), Springer: Springer Berlin · Zbl 1169.76001
[8] Sheretov, Yu. V., Continuum Dynamics under Spatiotemporal Averaging, (2009), RKhD: RKhD Moscow-Izhevsk, (in Russian) · Zbl 1357.76005
[9] Guermond, J.-L.; Popov, B., Viscous regularization of the euler equations and entropy principles, SIAM J. Appl. Math., 74, 2, 284-305, (2014) · Zbl 1446.76147
[10] Svärd, M., A new eulerian model for viscous and heat conducting compressible flows, Phys. A, 506, 350-375, (2018)
[11] Zlotnik, A.; Lomonosov, T., On conditions for weak conservativeness of regularized explicit finite-difference schemes for 1d barotropic gas dynamics equations, (Differential and Difference Equations with Applications. Differential and Difference Equations with Applications, Springer Proceedings in Mathematics & Statistics, vol. 230, (2018), Springer: Springer Cham), 635-647 · Zbl 1407.76106
[12] Zlotnik, A. A.; Lomonosov, T. A., On conditions for \(l^2\)-dissipativity of linearized explicit qgd finite-difference schemes for one-dimensional gas dynamics equations, Dokl. Math., 98, 2, 458-463, (2018) · Zbl 1408.76396
[13] Zlotnik, A. A., On conservative spatial discretizations of the barotropic quasi-gasdynamic system of equations with a potential body force, Comput. Math. Math. Phys., 56, 2, 303-319, (2016) · Zbl 1382.76198
[14] Gavrilin, V. A.; Zlotnik, A. A., 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) · Zbl 1330.76103
[15] Zlotnik, A.; Lomonosov, T., Verification of an entropy dissipative qgd-scheme for the 1d gas dynamics equations, Math. Model. Anal., 24, (2019), in press
[16] Zlotnik, A. A.; Chetverushkin, B. N., 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) · Zbl 1201.76232
[17] Elizarova, T. G.; Shil’nikov, E. V., Capabilities of a quasi-gasdynamic algorithm as applied to inviscid gas flow simulation, Comput. Math. Math. Phys., 49, 3, 532-548, (2009) · Zbl 1224.35331
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