Abstract
An asymptotic preserving and energy stable scheme for the barotropic Euler system under the low Mach number scaling is designed and analysed. A velocity shift proportional to the pressure gradient is introduced in the convective fluxes, which leads to the dissipation of mechanical energy and the entropy stability at all Mach numbers. The resolution of the semi-implicit in time and upwind finite volume in space fully-discrete scheme involves two steps: the solution of an elliptic problem for the density and an explicit evaluation for the velocity. The proposed scheme possesses several physically relevant attributes, such as the positivity of density, the entropy stability and the consistency with the weak formulation of the continuous Euler system. The AP property of the scheme, i.e. the boundedness of the mesh parameters with respect to the Mach number and its consistency with the incompressible limit system, is shown rigorously. The results of extensive case studies are presented to substantiate the robustness and efficiency of the proposed method.
Similar content being viewed by others
Data Availability
Enquiries about data availability should be directed to the authors.
References
Arun, K.R., Das Gupta, A.J., Samantaray, S.: Analysis of an asymptotic preserving low Mach number accurate IMEX-RK scheme for the wave equation system. Appl. Math. Comput. 411(20), 126469 (2021)
Arun, K.R., Krishnan, M., Samantaray, S.: A unified asymptotic preserving and well-balanced scheme for the Euler system with multiscale relaxation. Comput. Fluids 233(13), 105248 (2022)
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), 35 (2020). (32)
Bermúdez, A., Busto, S., Dumbser, M., Ferrín, J.L., Saavedra, L., Vázquez-Cendón, M.E.: A staggered semi-implicit hybrid FV/FE projection method for weakly compressible flows. J. Comput. Phys. 421(31), 109743 (2020)
Bijl, H., Wesseling, P.: A unified method for computing incompressible and compressible flows in boundary-fitted coordinates. J. Comput. Phys. 141(2), 153–173 (1998)
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)
Boscarino, S., Qiu, J.-M., Russo, G., Xiong, T.: A high order semi-implicit IMEX WENO scheme for the all-Mach isentropic Euler system. J. Comput. Phys. 392, 594–618 (2019)
Boscarino, S., Russo, G., Scandurra, L.: All Mach number second order semi-implicit scheme for the Euler equations of gas dynamics. J. Sci. Comput. 77(2), 850–884 (2018)
Busto, S., Río-Martín, L., Vázquez-Cendón, M.E., Dumbser, M.: A semi-implicit hybrid finite volume/finite element scheme for all Mach number flows on staggered unstructured meshes. Appl. Math. Comput. 402(29), 126117 (2021)
Busto, S., Tavelli, M., Boscheri, W., Dumbser, M.: Efficient high order accurate staggered semi-implicit discontinuous Galerkin methods for natural convection problems. Comput. Fluids 198(28), 104399 (2020)
Chalons, C., Girardin, M., Kokh, S.: An all-regime Lagrange-projection like scheme for the gas dynamics equations on unstructured meshes. Commun. Comput. Phys. 20(1), 188–233 (2016)
Chorin, A.J.: Numerical solution of the Navier-Stokes equations. Math. Comp. 22, 745–762 (1968)
Ciarlet, P.G.: Basic error estimates for elliptic problems. In: Handbook of numerical analysis, Vol. II, Handb. Numer. Anal., II, pages 17–351. North-Holland, Amsterdam, (1991)
Colella, P., Pao, K.: A projection method for low speed flows. J. Comput. Phys. 149(2), 245–269 (1999)
Couderc, F., Duran, A., Vila, J.-P.: An explicit asymptotic preserving low Froude scheme for the multilayer shallow water model with density stratification. J. Comput. Phys. 343, 235–270 (2017)
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)
Deimling, K.: Nonlinear functional analysis. Springer-Verlag, Berlin (1985)
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)
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)
Duran, A., Vila, J.-P., Baraille, R.: Semi-implicit staggered mesh scheme for the multi-layer shallow water system. C. R. Math. Acad. Sci. Paris 355(12), 1298–1306 (2017)
Duran, A., Vila, J.-P., Baraille, R.: Energy-stable staggered schemes for the Shallow Water equations. J. Comput. Phys. 401(24), 109051 (2020)
Ern, A., Guermond, J.-L.: Theory and practice of finite elements. Applied Mathematical Sciences, vol. 159. Springer-Verlag, New York (2004)
Eymard, R., Gallouët, T., Ghilani, M., Herbin, R.: Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes. IMA J. Numer. Anal. 18(4), 563–594 (1998)
Gallouët, T., Herbin, R., Latché, J.-C.: \(W^{1, q}\) stability of the Fortin operator for the MAC scheme. Calcolo 49(1), 63–71 (2012)
Gallouët, T., Herbin, R., Latché, J.-C.: On the weak consistency of finite volumes schemes for conservation laws on general meshes. SeMA J. 76(4), 581–594 (2019)
Gallouët, T., Herbin, R., Latché, J.-C.: Lax-Wendroff consistency of finite volume schemes for systems of non linear conservation laws: extension to staggered schemes. SeMA J. 79(2), 333–354 (2022)
Gallouët, T., Herbin, R., Latché, J.-C., Mallem, K.: Convergence of the marker-and-cell scheme for the incompressible Navier-Stokes equations on non-uniform grids. Found. Comput. Math. 18(1), 249–289 (2018)
Gallouët, T., Herbin, R., Maltese, D., Novotny, A.: Error estimates for a numerical approximation to the compressible barotropic Navier-Stokes equations. IMA J. Numer. Anal. 36(2), 543–592 (2016)
Gallouët, T., Maltese, D., Novotny, A.: Error estimates for the implicit MAC scheme for the compressible Navier-Stokes equations. Numer. Math. 141(2), 495–567 (2019)
Gastaldo, L., Herbin, R., Latché, J.-C., Therme, N.: A MUSCL-type segregated-explicit staggered scheme for the Euler equations. Comput. Fluids 175, 91–110 (2018)
Grenier, N., Vila, J.-P., Villedieu, P.: An accurate low-Mach scheme for a compressible two-fluid model applied to free-surface flows. J. Comput. Phys. 252, 1–19 (2013)
Guermond, J.-L.: Un résultat de convergence d’ordre deux en temps pour l’approximation des équations de Navier-Stokes par une technique de projection incrémentale. M2AN Math. Model. Numer. Anal., 33(1), 169–189 (1999)
Guermond, J.-L., Quartapelle, L.: Calculation of incompressible viscous flows by an unconditionally stable projection FEM. J. Comput. Phys. 132(1), 12–33 (1997)
Guillard, H., Viozat, C.: On the behaviour of upwind schemes in the low Mach number limit. Comput. Fluids 28(1), 63–86 (1999)
Haack, J., Jin, S., Liu, J.-G.: An all-speed asymptotic-preserving method for the isentropic Euler and Navier-Stokes equations. Commun. Comput. Phys. 12(4), 955–980 (2012)
Harlow, F.H., Amsden, A.A.: Numerical calculation of almost incompressible flow. J. Comput. Phys. 3(1), 80–93 (1968)
Harlow, F.H., Welch, J.E.: Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluids 8(12), 2182–2189 (1965)
Herbin, R., Kheriji, W., Latché, J.-C.: On some implicit and semi-implicit staggered schemes for the shallow water and Euler equations. ESAIM Math. Model. Numer. Anal. 48(6), 1807–1857 (2014)
Herbin, R., Latché, J.-C., Nasseri, Y., Therme, N.: A consistent quasi-second-order staggered scheme for the two-dimensional shallow water equations. IMA J. Numer. Anal. 43(1), 99–143 (2023)
Herbin, R., Latché, J.-C., Saleh, K.: Low Mach number limit of some staggered schemes for compressible barotropic flows. Math. Comp. 90(329), 1039–1087 (2021)
Issa, R.I., Gosman, A.D., Watkins, A.P.: The computation of compressible and incompressible recirculating flows by a noniterative implicit scheme. J. Comput. Phys. 62(1), 66–82 (1986)
Jin, S.: Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput. 21(2), 441–454 (1999)
Karki, K.C., Patankar, S.V.: Pressure based calculation procedure for viscous flows at all speedsin arbitrary configurations. AIAA J. 27(9), 1167–1174 (1989)
Klainerman, S., Majda, A.: Compressible and incompressible fluids. Comm. Pure Appl. Math. 35(5), 629–651 (1982)
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)
Kwatra, N., Su, J., Grétarsson, J.T., Fedkiw, R.: A method for avoiding the acoustic time step restriction in compressible flow. J. Comput. Phys. 228(11), 4146–4161 (2009)
Lions, P.-L., Masmoudi, N.: Incompressible limit for a viscous compressible fluid. J. Math. Pures Appl. 77(6), 585–627 (1998)
Moguen, Y., Kousksou, T., Bruel, P., Vierendeels, J., Dick, E.: Pressure-velocity coupling allowing acoustic calculation in low Mach number flow. J. Comput. Phys. 231(16), 5522–5541 (2012)
Munz, C.-D., Roller, S., Klein, R., Geratz, K.J.: The extension of incompressible flow solvers to the weakly compressible regime. Comput. Fluids 32(2), 173–196 (2003)
Nirenberg, L.: Topics in nonlinear functional analysis. In: Zehnder, E., Artino, R.A. (Eds) Chapter 6: Courant Lecture Notes in Mathematics. New York University, Courant Institute of Mathematical Sciences, vol 6, New York; American Mathematical Society, Providence, RI, (2001). Revised reprint of the 1974 original
Noelle, S., Bispen, G., Arun, K.R., Lukáčová-Medviová, 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)
O’Regan, D., Cho, Y.J., Chen, Y.-Q.: Topological degree theory and applications. Series in Mathematical Analysis and Applications, vol. 10. Chapman & Hall/CRC, Boca Raton, FL (2006)
Parisot, M., Vila, J.-P.: Centered-potential regularization for the advection upstream splitting method. SIAM J. Numer. Anal. 54(5), 3083–3104 (2016)
Park, J.H., Munz, C.-D.: Multiple pressure variables methods for fluid flow at all Mach numbers. Internat. J. Numer. Methods Fluids 49(8), 905–931 (2005)
Schochet, S.: Fast singular limits of hyperbolic PDEs. J. Differ. Equ. 114(2), 476–512 (1994)
Shin, D., Strikwerda, J.C.: Inf-sup conditions for finite-difference approximations of the Stokes equations. J. Aust. Math. Soc. Ser. B 39(1), 121–134 (1997)
Tavelli, M., Dumbser, M.: A pressure-based semi-implicit space–time discontinuous Galerkin method on staggered unstructured meshes for the solution of the compressible navier–stokes equations at all mach numbers. J. Comput. Phys. 341, 341–376 (2017)
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(25), 109723 (2020)
Toro, E.F.: Riemann solvers and numerical methods for fluid dynamics. Springer-Verlag, Berlin, 3rd edition. A practical introduction (2009)
Wall, C., Pierce, C.D., Moin, P.: A semi-implicit method for resolution of acoustic waves in low Mach number flows. J. Comput. Phys. 181(2), 545–563 (2002)
Weinan, E., Shu, C.W.: A numerical resolution study of high order essentially non-oscillatory schemes applied to incompressible flow. J. Comput. Phys. 110(1), 39–46 (1994)
Acknowledgements
The authors thank the anonymous reviewers for their comments and specific suggestions to improve the manuscript.
Funding
K.R.A. gratefully acknowledges Core Research Grant - CRG/2021/004078 from Science and Engineering Research Board, Department of Science & Technology, Government of India.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
K. R. A. gratefully acknowledges Core Research Grant - CRG/2021/004078 from Science and Engineering Research Board, Department of Science & Technology, Government of India.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Arun, K.R., Ghorai, R. & Kar, M. An Asymptotic Preserving and Energy Stable Scheme for the Barotropic Euler System in the Incompressible Limit. J Sci Comput 97, 73 (2023). https://doi.org/10.1007/s10915-023-02389-x
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10915-023-02389-x
Keywords
- Compressible Euler system
- Incompressible limit
- Asymptotic preserving
- Finite volume method
- MAC grid
- Entropy stability