## A unified asymptotic preserving and well-balanced scheme for the Euler system with multiscale relaxation.(English)Zbl 07464448

Summary: The design and analysis of a unified asymptotic preserving (AP) and well-balanced scheme for the Euler Equations with gravitational and frictional source terms is presented in this paper. The asymptotic behaviour of the Euler system in the limit of zero Mach and Froude numbers, and large friction is characterised by an additional scaling parameter. Depending on the values of this parameter, the Euler system relaxes towards a hyperbolic or a parabolic limit equation. Standard Implicit-Explicit Runge-Kutta schemes are incapable of switching between these asymptotic regimes. We propose a time semi-discretisation to obtain a unified scheme which is AP for the two different limits. A further reformulation of the semi-implicit scheme can be recast as a fully-explicit method in which the mass update contains both hyperbolic and parabolic fluxes. A space-time fully-discrete scheme is derived using a finite volume framework. A hydrostatic reconstruction strategy, an upwinding of the sources at the interfaces, and a careful choice of the central discretisation of the parabolic fluxes are used to achieve the well-balancing property for hydrostatic steady states. Results of several numerical case studies are presented to substantiate the theoretical claims and to verify the robustness of the scheme.

### MSC:

 35L45 Initial value problems for first-order hyperbolic systems 35L60 First-order nonlinear hyperbolic equations 35L65 Hyperbolic conservation laws 35L67 Shocks and singularities for hyperbolic equations 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs 76-XX Fluid mechanics
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### References:

 [1] Cercignani, C., (The Boltzmann Equation and its Applications. The Boltzmann Equation and its Applications, Applied Mathematical Sciences, vol. 67 (1988), Springer-Verlag: Springer-Verlag New York), xii+455 · Zbl 0646.76001 [2] Cercignani, C.; Illner, R.; Pulvirenti, M., (The Mathematical Theory of Dilute Gases. The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, vol. 106 (1994), Springer-Verlag: Springer-Verlag New York), viii+347 · Zbl 0813.76001 [3] Chapman, S.; Cowling, T. G., The Mathematical Theory of Non-Uniform Gases (1939), Cambridge University Press: Cambridge University Press Cambridge, xxiii+404 · Zbl 0063.00782 [4] Whitham, G. B., Linear and Nonlinear Waves, xvi+636 (1974), Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, Pure and Applied Mathematics · Zbl 0940.76002 [5] Chen, G. Q.; Levermore, C. D.; Liu, T.-P., Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm Pure Appl Math, 47, 6, 787-830 (1994) · Zbl 0806.35112 [6] Liu, T.-P., Hyperbolic conservation laws with relaxation, Comm Math Phys, 108, 1, 153-175 (1987) · Zbl 0633.35049 [7] Jin, S.; Xin, Z. P., The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm Pure Appl Math, 48, 3, 235-276 (1995) · Zbl 0826.65078 [8] Aregba-Driollet, D.; Natalini, R., Convergence of relaxation schemes for conservation laws, Appl Anal, 61, 1-2, 163-193 (1996) · Zbl 0887.65100 [9] Bouchut, F., Construction of BGK models with a family of kinetic entropies for a given system of conservation laws, J Stat Phys, 95, 1-2, 113-170 (1999) · Zbl 0957.82028 [10] Natalini, R., Convergence to equilibrium for the relaxation approximations of conservation laws, Comm Pure Appl Math, 49, 8, 795-823 (1996) · Zbl 0872.35064 [11] Natalini, R., Recent results on hyperbolic relaxation problems, (Analysis of Systems of Conservation Laws (Aachen, 1997). Analysis of Systems of Conservation Laws (Aachen, 1997), Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., vol. 99 (1999), Chapman & Hall/CRC, Boca Raton, FL), 128-198 · Zbl 0940.35127 [12] Boscarino, S.; Pareschi, L.; Russo, G., A unified IMEX Runge-Kutta approach for hyperbolic systems with multiscale relaxation, SIAM J Numer Anal, 55, 4, 2085-2109 (2017) · Zbl 1372.65261 [13] Jin, S., Runge-Kutta methods for hyperbolic conservation laws with stiff relaxation terms, J Comput Phys, 122, 1, 51-67 (1995) · Zbl 0840.65098 [14] Jin, S., Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J Sci Comput, 21, 2, 441-454 (1999) · Zbl 0947.82008 [15] Klainerman, S.; Majda, A., Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm Pure Appl Math, 34, 4, 481-524 (1981) · Zbl 0476.76068 [16] Jin, S., Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review, Riv. Math. Univ. Parma (N.S.), 3, 2, 177-216 (2012) · Zbl 1259.82079 [17] Boscarino, S.; Russo, G., Flux-explicit IMEX Runge-Kutta schemes for hyperbolic to parabolic relaxation problems, SIAM J Numer Anal, 51, 1, 163-190 (2013) · Zbl 1266.65141 [18] Caflisch, R. E.; Jin, S.; Russo, G., Uniformly accurate schemes for hyperbolic systems with relaxation, SIAM J Numer Anal, 34, 1, 246-281 (1997) · Zbl 0868.35070 [19] Dimarco, G.; Pareschi, L., Exponential Runge-Kutta methods for stiff kinetic equations, SIAM J Numer Anal, 49, 5, 2057-2077 (2011) · Zbl 1298.76150 [20] Jin, S.; Levermore, C. D., Numerical schemes for hyperbolic conservation laws with stiff relaxation terms, J Comput Phys, 126, 2, 449-467 (1996) · Zbl 0860.65089 [21] Jin, S.; Pareschi, L.; Toscani, G., Diffusive relaxation schemes for multiscale discrete-velocity kinetic equations, SIAM J Numer Anal, 35, 6, 2405-2439 (1998) · Zbl 0938.35097 [22] Jin, S.; Pareschi, L.; Toscani, G., Uniformly accurate diffusive relaxation schemes for multiscale transport equations, SIAM J Numer Anal, 38, 3, 913-936 (2000) · Zbl 0976.65091 [23] Klar, A., An asymptotic-induced scheme for nonstationary transport equations in the diffusive limit, SIAM J Numer Anal, 35, 3, 1073-1094 (1998) · Zbl 0918.65091 [24] Lemou, M.; Mieussens, L., A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit, SIAM J Sci Comput, 31, 1, 334-368 (2008) · Zbl 1187.82110 [25] Pareschi, L.; Russo, G., Implicit-Explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation, J Sci Comput, 25, 1-2, 129-155 (2005) · Zbl 1203.65111 [26] Albi, G.; Dimarco, G.; Pareschi, L., Implicit-explicit multistep methods for hyperbolic systems with multiscale relaxation, SIAM J Sci Comput, 42, 4, A2402-A2435 (2020) · Zbl 1455.65126 [27] Audusse, E.; Bouchut, F.; Bristeau, M.-O.; Klein, R.; Perthame, B., A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM J Sci Comput, 25, 6, 2050-2065 (2004) · Zbl 1133.65308 [28] Bouchut, F., (Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources. Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources, Frontiers in Mathematics (2004), Birkhäuser Verlag: Birkhäuser Verlag Basel), viii+135 · Zbl 1086.65091 [29] Chinnayya, A.; LeRoux, A.-Y.; Seguin, N., A well-balanced numerical scheme for the approximation of the shallow-water equations with topography: the resonance phenomenon, Int. J. Finite Vol., 1, 1, 33 (2004) [30] Fjordholm, U. S.; Mishra, S.; Tadmor, E., Well-balanced and energy stable schemes for the shallow water equations with discontinuous topography, J Comput Phys, 230, 14, 5587-5609 (2011) · Zbl 1452.35149 [31] Gallardo, J. M.; Parés, C.; Castro, M., On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas, J Comput Phys, 227, 1, 574-601 (2007) · Zbl 1126.76036 [32] Gosse, L., A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms, Comput Math Appl, 39, 9-10, 135-159 (2000) · Zbl 0963.65090 [33] Greenberg, J. M.; Leroux, A. Y., A well-balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM J Numer Anal, 33, 1, 1-16 (1996) · Zbl 0876.65064 [34] Kurganov, A.; Petrova, G., A second-order well-balanced positivity preserving central-upwind scheme for the saint-venant system, Commun Math Sci, 5, 1, 133-160 (2007) · Zbl 1226.76008 [35] Noelle, S.; Pankratz, N.; Puppo, G.; Natvig, J. R., Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows, J Comput Phys, 213, 2, 474-499 (2006) · Zbl 1088.76037 [36] Noelle, S.; Xing, Y.; Shu, C.-W., High-order well-balanced finite volume WENO schemes for shallow water equation with moving water, J Comput Phys, 226, 1, 29-58 (2007) · Zbl 1120.76046 [37] Xing, Y.; Shu, C.-W., High order finite difference WENO schemes with the exact conservation property for the shallow water equations, J Comput Phys, 208, 1, 206-227 (2005) · Zbl 1114.76340 [38] Xing, Y.; Shu, C.-W., High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms, J Comput Phys, 214, 2, 567-598 (2006) · Zbl 1089.65091 [39] Xu, K., A well-balanced gas-kinetic scheme for the shallow-water equations with source terms, J Comput Phys, 178, 2, 533-562 (2002) · Zbl 1017.76071 [40] Arun, K. R.; Samantaray, S., Asymptotic preserving low mach number accurate IMEX finite volume schemes for the isentropic Euler equations, J Sci Comput, 82, 2 (2020), Art. 35, 32 · Zbl 1434.76077 [41] Bouchut, F.; Franck, E.; Navoret, L., A low cost semi-implicit low-mach relaxation scheme for the full Euler equations, J Sci Comput, 83, 1 (2020), Paper No. 24, 47 · Zbl 1434.76078 [42] Dimarco, G.; Loubère, R.; Vignal, M.-H., Study of a new asymptotic preserving scheme for the Euler system in the low mach number limit, SIAM J Sci Comput, 39, 5, A2099-A2128 (2017) · Zbl 1391.76401 [43] 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 [44] Noelle, S.; Bispen, G.; Arun, K. R.; Lukáčová-Medviďová, M.; Munz, C.-D., A weakly asymptotic preserving low mach number scheme for the Euler equations of gas dynamics, SIAM J Sci Comput, 36, 6, B989-B1024 (2014) · Zbl 1321.76053 [45] Bispen, G.; Arun, K. R.; Lukáčová-Medvid’ová, 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 [46] Bispen, G.; Lukáčová-Medviďová, M.; Yelash, L., Asymptotic preserving IMEX finite volume schemes for low mach number Euler equations with gravitation, J Comput Phys, 335, 222-248 (2017) · Zbl 1375.76023 [47] Thomann, A.; Puppo, G.; Klingenberg, C., An all speed second order well-balanced IMEX relaxation scheme for the Euler equations with gravity, J Comput Phys, 420, 109723, 25 (2020) [48] Natalini, R.; Ribot, M.; Twarogowska, M., A well-balanced numerical scheme for a one dimensional quasilinear hyperbolic model of chemotaxis, Commun Math Sci, 12, 1, 13-39 (2014) · Zbl 1310.92008 [49] Natalini, R.; Ribot, M.; Twarogowska, M., A numerical comparison between degenerate parabolic and quasilinear hyperbolic models of cell movements under chemotaxis, J Sci Comput, 63, 3, 654-677 (2015) · Zbl 1334.92064 [50] Perthame, B.; Simeoni, C., Convergence of the upwind interface source method for hyperbolic conservation laws, (Hyperbolic Problems: Theory, Numerics, Applications (2003), Springer, Berlin), 61-78 · Zbl 1064.65098 [51] Varma, D.; Chandrashekar, P., A second-order, discretely well-balanced finite volume scheme for Euler equations with gravity, Comput. Fluids, 181, 292-313 (2019) · Zbl 1410.76270 [52] Pareschi, L.; Russo, G., Implicit-explicit runge-kutta schemes for stiff systems of differential equations, (Recent Trends in Numerical Analysis. Recent Trends in Numerical Analysis, Adv. Theory Comput. Math., vol. 3 (2001), Nova Sci. Publ., Huntington, NY), 269-288 · Zbl 1018.65093 [53] 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
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