March  2012, 5(1): 155-184. doi: 10.3934/krm.2012.5.155

Second order all speed method for the isentropic Euler equations

1. 

Department of Mathematics and Institute of Natural Sciences, Shanghai Jiao Tong University, No. 800 Dong Chuan Road, Minhang, Shanghai 200240, China

Received  September 2011 Revised  September 2011 Published  January 2012

Standard hyperbolic solvers for the compressible Euler equations cause increasing approximation errors and have severe stability requirement in the low Mach number regime. It is desired to design numerical schemes that are suitable for all Mach numbers. A second order in both space and time all speed method is developed in this paper, which is an improvement of the semi-implicit framework proposed in [5].
    The second order time discretization is based on second order Runge-Kutta method combined with Crank-Nicolson with some implicit terms. This semi-discrete framework is crucial to obtain second order convergence, as well as maintain the asymptotic preserving (AP) property. The AP property indicates that the right limit can be captured in the low Mach number regime. For the space discretization, the pressure term in the momentum equation is divided into two parts. Two subsystems are formed correspondingly, each using different space discretizations. One is discretized by Kurganov-Tadmor central scheme (KT), while the other one is reformulated into an elliptic equation. The proper subsystem division varies with time and the scheme becomes explicit when the time step is small enough.
    Compared with previous semi-implicit method, this framework is simpler and natural, with only two linear elliptic equations needed to be solved for each time step. It maintains the AP property of the first order method in [5], improves accuracy and reduces the diffusivity significantly.
Citation: Min Tang. Second order all speed method for the isentropic Euler equations. Kinetic and Related Models, 2012, 5 (1) : 155-184. doi: 10.3934/krm.2012.5.155
References:
[1]

M. P. Bonner, "Compressible Subsonic Flow on a Staggered Grid," Master thesis, The University of British Columbia, 2007.

[2]

F. Cordier, P. Degond and A. Kumbaro, An asymptotic preserving scheme for the low Mach number limit of the Navier Stokes equations,, preprint., (). 

[3]

P. Degond, S. Jin and J.-G. Liu, Mach-number uniform asymptotic-preserving gauge schemes for compressible flows, Bulletin of the Institute of Mathematics, Academia Sinica, New Series, 2 (2007), 851-892.

[4]

P. Degond, P.-F. Peyrard, G. Russo and P. Villedieu, Polynomial upwind schemes for hyperbolic systems, C. R. Acad.Sci. Paris Sér. I Math., 328 (1999), 479-483. doi: 10.1016/S0764-4442(99)80194-3.

[5]

P. Degond and M. Tang, All speed scheme for the low Mach number limit of the isentropic Euler equations, Communications in Computational Physics, 10 (2011), 1-31.

[6]

P. M. Gresho, On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. I: Theory, Computational Methods in Flow Analysis (Okayama, 1988), Int. J. Numer. Methods Fluids, 11 (1990), 587-620. doi: 10.1002/fld.1650110509.

[7]

P. M. Gresho and S. T. Chan, On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. II: Implementation, Computational methods in flow analysis (Okayama, 1988), Int. J. Numer. Methods Fluids, 11 (1990), 621-659. doi: 10.1002/fld.1650110510.

[8]

F. Golse, S. Jin and C. D. Levermore, The convergence of numerical transfer schemes in diffusive regimes I: Discrete-ordinate method, SIAM J. Numer. Anal., 36 (1999), 1333-1369. doi: 10.1137/S0036142997315986.

[9]

S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comp., 21 (1999), 441-454. doi: 10.1137/S1064827598334599.

[10]

J. R. Haack and C. D. Hauck, Oscillatory behavior of asymptotic-preserving splitting methods for a linear model of diffusive relaxation, Kinetic and Related Models, 1 (2008), 573-590.

[11]

J. Haack, S. Jin and J. G. Liu, An all-speed asymptotic-preserving method for the isentropic Euler and Navier-Stokes equation,, preprint., (). 

[12]

F. H. Harlow and A. Amsden, A numerical fluid dynamics calculation method for all flow speeds, J. Comput. Phys, 8 (1971), 197-213. doi: 10.1016/0021-9991(71)90002-7.

[13]

F. H. Harlow and J. E. Welch, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluid, 8 (1965), 2182-2189.

[14]

D. R. van der Heul, C. Vuik and P. Wesseling, A conservative pressure-correction method for flow at all speeds, Compt. & Fluids, 32 (2003), 1113-1132.

[15]

R. I. Issa, A. D. Gosman and A. P. Watkins, The computation of compressible and incompressible flow of fluid with a free surface, Phys. Fluids, 8 (1965), 2182-2189.

[16]

S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Communication on Pure and Applied Mathematics, 34 (1981), 481-524. doi: 10.1002/cpa.3160340405.

[17]

S. Klainerman and A. Majda, Compressible and incompressible fluids, Communication on Pure and Applied Mathematics, 35 (1982), 629-651. doi: 10.1002/cpa.3160350503.

[18]

R. Klein, N. Botta, T. Schneider, C. D. Munz, S. Roller, A. Meister, L. Hoffmann and T. Sonar, Asymptotic adaptive methods for multi-scale problems in fluid mechanics, J. Eng. Math., 39 (2001), 261-343. doi: 10.1023/A:1004844002437.

[19]

A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comp. Phys., 160 (2000), 214-282.

[20]

A. Kurganov and E. Tadmor, Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers, Numerical Methods for Partial Differential Equations, 18 (2002), 548-608.

[21]

A. Kurganov and D. Levy, A third-order semidiscrete central scheme for conservation laws and convection-diffusion equations, SIAM J. Sci. Comput., 22 (2000), 1461-1488. doi: 10.1137/S1064827599360236.

[22]

A. Kurganov, S. Noelle and G. Petrova, Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations, SIAM J. Sci. Comp., 23 (2001), 707-740. doi: 10.1137/S1064827500373413.

[23]

R. J. LeVeque, "Numerical Methods for Conservation Laws," Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1992.

[24]

C.-D. Munz, S. Roller, R. Klein and K. J. Geratz, The extension of incompressible flow solvers to the weakly compressible regime, Comp. Fluids, 32 (2003), 173-196. doi: 10.1016/S0045-7930(02)00010-5.

[25]

J. H. Park and C.-D. Munz, Multiple pressure variables methods for fluid flow at all Mach numbers, Int. J. Numer. Meth. Fluid, 49 (2005), 905-931. doi: 10.1002/fld.1032.

[26]

S. V. Patankar, "Numerical Heat Transfer and Fluid Flow," McGraw-Hill, New York, 1980.

[27]

R. Klein, Semi-implicit extension of a Godunov-type scheme based on low Mach number asymptotics. I: One-dimensional flow, J. Comp. Phys., 121 (1995), 213-237. doi: 10.1016/S0021-9991(95)90034-9.

[28]

N. Kwatra, J. Su, J. T. Grétarsson and R. Fedkiw, A method for avoiding the acoustic time step restriction in compressible flow, J. Comp. Phys., 228 (2009), 4146-4161. doi: 10.1016/j.jcp.2009.02.027.

[29]

K. Nerinckx, J. Vierendeels and E. Dick, A Mach-uniform algorithm: Coupled versus segregated approach, J. Comp. Phys., 224 (2007), 314-331. doi: 10.1016/j.jcp.2007.02.008.

[30]

F. Rieper and G. Bader, The influence of cell geometry on the accuracy of upwind schemes in the low Mach number regime, J. Comp. Phys., 228 (2009), 2918-2933. doi: 10.1016/j.jcp.2009.01.002.

show all references

References:
[1]

M. P. Bonner, "Compressible Subsonic Flow on a Staggered Grid," Master thesis, The University of British Columbia, 2007.

[2]

F. Cordier, P. Degond and A. Kumbaro, An asymptotic preserving scheme for the low Mach number limit of the Navier Stokes equations,, preprint., (). 

[3]

P. Degond, S. Jin and J.-G. Liu, Mach-number uniform asymptotic-preserving gauge schemes for compressible flows, Bulletin of the Institute of Mathematics, Academia Sinica, New Series, 2 (2007), 851-892.

[4]

P. Degond, P.-F. Peyrard, G. Russo and P. Villedieu, Polynomial upwind schemes for hyperbolic systems, C. R. Acad.Sci. Paris Sér. I Math., 328 (1999), 479-483. doi: 10.1016/S0764-4442(99)80194-3.

[5]

P. Degond and M. Tang, All speed scheme for the low Mach number limit of the isentropic Euler equations, Communications in Computational Physics, 10 (2011), 1-31.

[6]

P. M. Gresho, On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. I: Theory, Computational Methods in Flow Analysis (Okayama, 1988), Int. J. Numer. Methods Fluids, 11 (1990), 587-620. doi: 10.1002/fld.1650110509.

[7]

P. M. Gresho and S. T. Chan, On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. II: Implementation, Computational methods in flow analysis (Okayama, 1988), Int. J. Numer. Methods Fluids, 11 (1990), 621-659. doi: 10.1002/fld.1650110510.

[8]

F. Golse, S. Jin and C. D. Levermore, The convergence of numerical transfer schemes in diffusive regimes I: Discrete-ordinate method, SIAM J. Numer. Anal., 36 (1999), 1333-1369. doi: 10.1137/S0036142997315986.

[9]

S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comp., 21 (1999), 441-454. doi: 10.1137/S1064827598334599.

[10]

J. R. Haack and C. D. Hauck, Oscillatory behavior of asymptotic-preserving splitting methods for a linear model of diffusive relaxation, Kinetic and Related Models, 1 (2008), 573-590.

[11]

J. Haack, S. Jin and J. G. Liu, An all-speed asymptotic-preserving method for the isentropic Euler and Navier-Stokes equation,, preprint., (). 

[12]

F. H. Harlow and A. Amsden, A numerical fluid dynamics calculation method for all flow speeds, J. Comput. Phys, 8 (1971), 197-213. doi: 10.1016/0021-9991(71)90002-7.

[13]

F. H. Harlow and J. E. Welch, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluid, 8 (1965), 2182-2189.

[14]

D. R. van der Heul, C. Vuik and P. Wesseling, A conservative pressure-correction method for flow at all speeds, Compt. & Fluids, 32 (2003), 1113-1132.

[15]

R. I. Issa, A. D. Gosman and A. P. Watkins, The computation of compressible and incompressible flow of fluid with a free surface, Phys. Fluids, 8 (1965), 2182-2189.

[16]

S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Communication on Pure and Applied Mathematics, 34 (1981), 481-524. doi: 10.1002/cpa.3160340405.

[17]

S. Klainerman and A. Majda, Compressible and incompressible fluids, Communication on Pure and Applied Mathematics, 35 (1982), 629-651. doi: 10.1002/cpa.3160350503.

[18]

R. Klein, N. Botta, T. Schneider, C. D. Munz, S. Roller, A. Meister, L. Hoffmann and T. Sonar, Asymptotic adaptive methods for multi-scale problems in fluid mechanics, J. Eng. Math., 39 (2001), 261-343. doi: 10.1023/A:1004844002437.

[19]

A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comp. Phys., 160 (2000), 214-282.

[20]

A. Kurganov and E. Tadmor, Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers, Numerical Methods for Partial Differential Equations, 18 (2002), 548-608.

[21]

A. Kurganov and D. Levy, A third-order semidiscrete central scheme for conservation laws and convection-diffusion equations, SIAM J. Sci. Comput., 22 (2000), 1461-1488. doi: 10.1137/S1064827599360236.

[22]

A. Kurganov, S. Noelle and G. Petrova, Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations, SIAM J. Sci. Comp., 23 (2001), 707-740. doi: 10.1137/S1064827500373413.

[23]

R. J. LeVeque, "Numerical Methods for Conservation Laws," Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1992.

[24]

C.-D. Munz, S. Roller, R. Klein and K. J. Geratz, The extension of incompressible flow solvers to the weakly compressible regime, Comp. Fluids, 32 (2003), 173-196. doi: 10.1016/S0045-7930(02)00010-5.

[25]

J. H. Park and C.-D. Munz, Multiple pressure variables methods for fluid flow at all Mach numbers, Int. J. Numer. Meth. Fluid, 49 (2005), 905-931. doi: 10.1002/fld.1032.

[26]

S. V. Patankar, "Numerical Heat Transfer and Fluid Flow," McGraw-Hill, New York, 1980.

[27]

R. Klein, Semi-implicit extension of a Godunov-type scheme based on low Mach number asymptotics. I: One-dimensional flow, J. Comp. Phys., 121 (1995), 213-237. doi: 10.1016/S0021-9991(95)90034-9.

[28]

N. Kwatra, J. Su, J. T. Grétarsson and R. Fedkiw, A method for avoiding the acoustic time step restriction in compressible flow, J. Comp. Phys., 228 (2009), 4146-4161. doi: 10.1016/j.jcp.2009.02.027.

[29]

K. Nerinckx, J. Vierendeels and E. Dick, A Mach-uniform algorithm: Coupled versus segregated approach, J. Comp. Phys., 224 (2007), 314-331. doi: 10.1016/j.jcp.2007.02.008.

[30]

F. Rieper and G. Bader, The influence of cell geometry on the accuracy of upwind schemes in the low Mach number regime, J. Comp. Phys., 228 (2009), 2918-2933. doi: 10.1016/j.jcp.2009.01.002.

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