Skip to main content Accessibility help

An All-Speed Asymptotic-Preserving Method for the Isentropic Euler and Navier-Stokes Equations

  • Jeffrey Haack (a1) (a2), Shi Jin (a1) and Jian‐Guo Liu (a3)


The computation of compressible flows becomes more challenging when the Mach number has different orders of magnitude. When the Mach number is of order one, modern shock capturing methods are able to capture shocks and other complex structures with high numerical resolutions. However, if the Mach number is small, the acoustic waves lead to stiffness in time and excessively large numerical viscosity, thus demanding much smaller time step and mesh size than normally needed for incompressible flow simulation. In this paper, we develop an all-speed asymptotic preserving (AP) numerical scheme for the compressible isentropic Euler and Navier-Stokes equations that is uniformly stable and accurate for all Mach numbers. Our idea is to split the system into two parts: one involves a slow, nonlinear and conservative hyperbolic system adequate for the use of modern shock capturing methods and the other a linear hyperbolic system which contains the stiff acoustic dynamics, to be solved implicitly. This implicit part is reformulated into a standard pressure Poisson projection system and thus possesses sufficient structure for efficient fast Fourier transform solution techniques. In the zero Mach number limit, the scheme automatically becomes a projection method-like incompressible solver. We present numerical results in one and two dimensions in both compressible and incompressible regimes.


Corresponding author



Hide All
[1]Abarbanel, S., Duth, P. and Gottlieb, D., Splitting methods for low Mach number Euler and Navier-Stokes equations, Comput. Fluids, 17 (1989), 1–12.
[2]Bradford, B., The fast staggered transform, composite symmetries and compact symmetric algorithms, IBM J. Res. Develop, 38 (1994), 117–129.
[3]Chorin, A., Numerical solution of the Navier-Stokes equations, Math. Comput., 22 (1968), 745–762.
[4]Chorin, A., A numerical method for solving incompressible viscous flow problems, J. Comput. Phys., 2 (1967), 12–26.
[5]Colella, P. and Pao, K., A projection method for low speed flows, J. Comput. Phys., 149 (1999), 245–269.
[6]Crispel, P., Degond, P. and Vignal, M.-H., An asymptotic preserving scheme for the two-fluid Euler-Poisson model in the quasineutral limit, J. Comput. Phys., 223 (2007), 208–234.
[7]Degond, P., Jin, S. and Liu, J.-G., Mach-number uniform asymptotic-preserving gauge schemes for compressible flows, Bulletin of the Institute of Mathematics, Academia Sinica, New Series, 2(4) (2007), 851–892.
[8]Degond, P. and Tang, M., All speed scheme for the low Mach number limit of the isentropic Euler equation, Commun. Comput. Phys., 10 (2011), 1–31.
[9]Degond, P., Liu, J.-G. and Vignal, M.-H., Analysis of an asymptotic preserving scheme for the Euler-Poisson system in the quasineutral limit, SIAM J. Num. Anal., 46 (2008), 1298–1322.
[10] W. E and Liu, J.-G., Vorticity boundary condition and related issues for finite difference schemes, J. Comput. Phys., 124 (1996), 368–382.
[11]Gatti-Bono, C. and Colella, P., An analastic allspeed projection method for gravitationally stratified flows, J. Comput. Phys., 216 (2006), 589–615.
[12]Golse, F., Jin, S. and Levermore, C. D., The convergence of numerical transfer schemes in diffusive regimes I: the discrete-ordinate method, SIAM J. Numer. Anal., 36 (1999), 1333–1369.
[13]Guillard, H. and Viozat, C., On the behaviour of upwind schemes in the low Mach number limit, Comput. Fluids, 28 (1998), 63–86.
[14]Gustaffson, B. and Stoor, H., Navier-Stokes equations for almost incompressible flow, SIAM J. Numer. Anal., 28 (1991), 441–454.
[15]Harlow, F. H. and Amsden, A., A numerical fluid dynamics calculation method for all flow speeds, J. Comput. Phys., 8 (1971), 179–213.
[16]Harlow, F. H. and Welch, J. E., Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluid, 8 (1965), 2182–2189.
[17]Jin, S., Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput., 21 (1999), 441–454.
[18]Jin, S., Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review. Lecture Notes for Summer School on “Methods and Models of Kinetic Theory” (M&MKT), Porto Ercole (Grosseto, Italy), June 2010. To appear in Rivista di Matematica della Universita di Parma.
[19]Jin, S. and Levermore, C. D., The discrete-ordinate method in diffusive regime, Trans. Theory. Stat. Phys., 20 (1991), 413–439.
[20]Kadioglu, S. Y. and Sussman, M., Adaptive solution tecniques for simulating underwater explosions and implosions, J. Comput. Phys., 227 (2008), 2083–2104.
[21]Kadioglu, S. Y., Sussman, M., Osher, S., Wright, J. P. and Kang, M., A second order primitive preconditioner for solving all speed multi-phase flows, J. Comput. Phys., 209 (2005), 477–503.
[22]Kadioglu, S. Y., Klein, R. and Minion, M. L., A fourth order auxilary variable projection method for zero‒Mach number gas dynamics, J. Comput. Phys., 277 (2008), 2012–2043.
[23]Klainerman, S. and Majda, A., Singular limits of quasilinear hyperbolic systems with large parametersand the incompressible limit of compressible fluids, Commun. Pure Appl. Math., 34 (1981), 481–524.
[24]Klein, R., Semi-implicit extension of a Godunov-type scheme based on low Mach number asymptotics I: one dimensional flow, J. Comput. Phys., 121 (1995), 213237.
[25]Kurganov, A. and Tadmor, E., New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys., 160 (2000), 241–282.
[26]Larsen, E. W. and Morel, J. E., Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes II, J. Comput. Phys., 83 (1989), 212–236.
[27]Lax, P. D. and Liu, X.-D., Solution of two-dimensional Riemann problems of gas dynamics by positive schemes, SIAM J. Sci. Comput., 19 (1998), 319340.
[28]Liu, J.-G., Liu, J. and Pego, R. L., Stable and accurate pressure approximation for unsteady incompressible viscous flow, J. Comput. Phys., 229 (2010), 3428–3453.
[29]Swarztrauber, P., Fast Poisson solvers, in Studies in Numerical Analysis, ed. G. Golub, AMS, 1984.
[30]Temam, R., Sur l’approximation de la solution des equations de Navier-Stokes par la méthode des fractionnarires II, Arch. Rat. Mech. Anal., 33 (1969), 377–385.
[31]Turkel, E., Preconditioned methods for solving the incompressible and low speed compressible equations, J. Comput. Phys., 72 (1987), 189–209.



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed