zbMATH — the first resource for mathematics

An all Mach number relaxation upwind scheme. (English) Zbl 1447.65043
The authors deal with the low Mach number approximation of flows governed by the compressible Euler equations. They use the standard Suliciu relaxation technique and showed that it is not efficient for the approximation of these flow regimes. Instead, a modified relaxation scheme is constructed. It is shown that the numerical diffusion of the upwind scheme is controlled in the low Mach number case and that the relaxation scheme is robust. The asymptotic preserving property is also shown. Some numerical tests are given in 1D and 2D cases to demonstrate the performance of the modified relaxation scheme compared to a standard scheme.
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35Q31 Euler equations
Full Text: DOI
[1] Barsukow, W.; Edelmann, P.; Klingenberg, C.; Miczek, F.; Röpke, F., A numerical scheme for the compressible low-Mach number regime of idela fluid dynamics, J. Sci. Comput., 72, 2, 623-646 (2017) · Zbl 1459.65166
[2] Baudin, M.; Berthon, C.; Coquel, F.; Masson, R.; Tran, Q. H., A relaxation method for two-phase flow models with hydrodynamic closure law, Numer. Math., 99, 3, 411-440 (2005) · Zbl 1204.76025
[3] Bispen, G.; Arun, K. R.; Lukáčová-Medvidóvá, M.; Noelle, S., IMEX large time step finite volume methods for low Froude number shallow water flows, Commun. Comput. Phys., 16, 2, 307-347 (2014) · Zbl 1373.76117
[4] Boscarino, S.; Pareschi, L.; Russo, G., Implicit-Explicit Runge-Kutta Schemes for Hyperbolic Systems and Kinetic Equations in the Diffusion Limit, SIAM J. Sci. Comput., 35 (2013) · Zbl 1264.65150
[5] Bouchut, F., Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources (2004), Birkhäuser · Zbl 1086.65091
[6] Chalons, C.; Coquel, F.; Godlewski, E.; Raviart, P.-A.; Seguin, N., Godunov-type schemes for hyperbolic systems with parameter-dependent source. The case of Euler system with friction, Math. Models Methods Appl. Sci., 20, 11, 2109-2166 (2010) · Zbl 1213.35034
[7] Chen, G. Q.; Levermore, C. D.; Liu, T.-P., Hyperbolic conservation laws with stiff relaxation terms and entropy, Commun. Pure Appl. Math., 47, 787-830 (1994) · Zbl 0806.35112
[8] Coquel, F.; Perthame, B., Relaxation of Energy and Approximate Riemann Solvers for General Pressure Laws in Fluid Dynamics, SIAM J. Numer. Anal., 35, 6, 2223-2249 (1998) · Zbl 0960.76051
[9] De Coninck, A.; De Baets, B.; Kourounis, D.; Verbosio, F.; Schenk, O.; Maenhout, S. n.; Fostier, J., Needles: Toward Large-Scale Genomic Prediction with Marker-by-Environment Interaction, Genetics, 203, 1, 543-555 (2016)
[10] Degond, P.; Tang, M., All Speed Scheme for the Low Mach Number Limit of the Isentropic Euler Equations, Commun. Comput. Phys., 10, 1, 1-31 (2011) · Zbl 1364.76129
[11] Dellacherie, S., Analysis of Godunov type schemes applied to the compressible Euler system at low Mach number, J. Comput. Phys., 229, 4, 978-1016 (2010) · Zbl 1329.76228
[12] Dimarco, G.; Pareschi, L., Asymptotic Preserving Implicit-Explicit Runge-Kutta Methods for Nonlinear Kinetic Equations, SIAM J. Numer. Anal., 51 (2013) · Zbl 1268.76055
[13] Feireisl, E.; Klingenberg, C.; Markfelder, S., On the low Mach number limit for the compressible Euler system · Zbl 1420.35209
[14] Godlewski, E.; Raviart, P.-A., Numerical approximation of hyperbolic systems of conservation laws, 118 (1996), Springer · Zbl 1063.65080
[15] Guillard, H.; Murrone, A., On the behavior of upwind schemes in the low Mach number limit: II. Godunov type schemes, Computers & Fluids, 33, 655-675 (2004) · Zbl 1049.76040
[16] Guillard, H.; Viozat, C., On the behaviour of upwind schemes in the low Mach number limit, Computers & Fluids, 28, 63-86 (1999) · Zbl 0963.76062
[17] Haack, J.; Jin, S.; Liu, J., An all-speed asymptotic-preserving method for the isentropic Euler and Navier-Stokes equations, Commun. Comput. Phys., 12, 4, 955-980 (2012) · Zbl 1373.76284
[18] Harten, A.; Lax, P. D.; Van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25, 35-61 (1983) · Zbl 0565.65051
[19] Jin, S., Efficient Asymptotic-Preserving (AP) Schemes For Some Multiscale Kinetic Equations, SIAM J. Sci. Comput., 21, 2, 441-454 (1999) · Zbl 0947.82008
[20] Kennedy, C. A.; Carpenter, M. H., Additive Runge-Kutta Schemes for Convection-Diffusion-Reaction Equation (2001)
[21] Klainerman, S.; Majda, A., Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Commun. Pure Appl. Math., 34, 4, 481-524 (1981) · Zbl 0476.76068
[22] Klein, R., Semi-implicit extension of a godunov-type scheme based on low mach number asymptotics I: One-dimensional flow, J. Comput. Phys., 121, 2, 213-237 (1995) · Zbl 0842.76053
[23] Klein, R.; Botta, N.; Schneider, T.; Munz, C. D.; Roller, S.; Meister, A.; Hoffmann, L.; Sonar, T., Asymptotic adaptive methods for multi-scale problems in fluid mechanics, J. Eng. Math., 39, 1, 261-343 (2001) · Zbl 1015.76071
[24] Kourounis, D.; Fuchs, A.; Schenk, O., Towards the Next Generation of Multiperiod Optimal Power Flow Solvers, IEEE Trans. Power Syst., 33, 4, 4005-4014 (2018)
[25] Miczek, F., Simulation of low Mach number astrophysical flows (2013)
[26] Miczek, F.; Röpke, F.; Edelmann, P., A new numerical solver for flows at various Mach numbers, Astron. Astrophys., 576, A50, 16 p. pp. (2015)
[27] Noelle, S.; Bispen, G.; Arun, K. R.; Lukáčová-Medvidóvá, M.; Munz, C. D., An Asymptotic Preserving all Mach Number Scheme for the Euler Equations of Gas Dynamics, SIAM J. Sci. Comput., 36, 6, 989-1024 (2014) · Zbl 1321.76053
[28] Perthame, B.; Shu, C. W., On positivity preserving finite volume schemes for Euler equations, Numer. Math., 73, 1, 119-130 (1996) · Zbl 0857.76062
[29] Roe, P. L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., 43, 2, 357-372 (1981) · Zbl 0474.65066
[30] Sod, G. A., A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys., 27, 1, 1-31 (1978) · Zbl 0387.76063
[31] Suliciu, I., On modelling phase transitions by means of rate-type constitutive equations, shock wave structure, Int. J. Eng. Sci., 28, 829-841 (1990) · Zbl 0738.73007
[32] Suliciu, I., Some stability-instability problems in phase transitions modelled by piecewise linear elastic or viscoelastic constitutive equations, Int. J. Eng. Sci., 30, 483-494 (1992) · Zbl 0752.73009
[33] Turkel, E., Preconditioned Methods for Solving the Incompressible and Low Speed Compressible Equations, J. Comput. Phys., 72 (1987) · Zbl 0633.76069
[34] Verbosio, F.; De Coninck, A.; Kourounis, D.; Schenk, O., Enhancing the scalability of selected inversion factorization algorithms in genomic prediction, J. Comput. Sci., 22, Supplement C, 99-108 (2017)
[35] Weiss, J. M.; Smith, W. A., Preconditioning applied to variable and constant density flows, AIAA J., 33, 11, 2050-2057 (1995) · Zbl 0849.76072
[36] Whitham, G. B., Linear and nonlinear waves, xvi+636 p. pp. (1974), John Wiley & Sons · Zbl 0373.76001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.