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 & 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, (2007). Google Scholar

[2]

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

[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, 2 (2007), 851. Google Scholar

[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. doi: 10.1016/S0764-4442(99)80194-3. Google Scholar

[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. Google Scholar

[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, 11 (1990), 587. doi: 10.1002/fld.1650110509. Google Scholar

[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, 11 (1990), 621. doi: 10.1002/fld.1650110510. Google Scholar

[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. doi: 10.1137/S0036142997315986. Google Scholar

[9]

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

[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. Google Scholar

[11]

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

[12]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. doi: 10.1002/cpa.3160340405. Google Scholar

[17]

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

[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. doi: 10.1023/A:1004844002437. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. doi: 10.1137/S1064827599360236. Google Scholar

[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. doi: 10.1137/S1064827500373413. Google Scholar

[23]

R. J. LeVeque, "Numerical Methods for Conservation Laws,", Second edition, (1992). Google Scholar

[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. doi: 10.1016/S0045-7930(02)00010-5. Google Scholar

[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. doi: 10.1002/fld.1032. Google Scholar

[26]

S. V. Patankar, "Numerical Heat Transfer and Fluid Flow,", McGraw-Hill, (1980). Google Scholar

[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. doi: 10.1016/S0021-9991(95)90034-9. Google Scholar

[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. doi: 10.1016/j.jcp.2009.02.027. Google Scholar

[29]

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

[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. doi: 10.1016/j.jcp.2009.01.002. Google Scholar

show all references

References:
[1]

M. P. Bonner, "Compressible Subsonic Flow on a Staggered Grid,", Master thesis, (2007). Google Scholar

[2]

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

[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, 2 (2007), 851. Google Scholar

[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. doi: 10.1016/S0764-4442(99)80194-3. Google Scholar

[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. Google Scholar

[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, 11 (1990), 587. doi: 10.1002/fld.1650110509. Google Scholar

[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, 11 (1990), 621. doi: 10.1002/fld.1650110510. Google Scholar

[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. doi: 10.1137/S0036142997315986. Google Scholar

[9]

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

[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. Google Scholar

[11]

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

[12]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. doi: 10.1002/cpa.3160340405. Google Scholar

[17]

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

[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. doi: 10.1023/A:1004844002437. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. doi: 10.1137/S1064827599360236. Google Scholar

[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. doi: 10.1137/S1064827500373413. Google Scholar

[23]

R. J. LeVeque, "Numerical Methods for Conservation Laws,", Second edition, (1992). Google Scholar

[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. doi: 10.1016/S0045-7930(02)00010-5. Google Scholar

[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. doi: 10.1002/fld.1032. Google Scholar

[26]

S. V. Patankar, "Numerical Heat Transfer and Fluid Flow,", McGraw-Hill, (1980). Google Scholar

[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. doi: 10.1016/S0021-9991(95)90034-9. Google Scholar

[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. doi: 10.1016/j.jcp.2009.02.027. Google Scholar

[29]

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

[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. doi: 10.1016/j.jcp.2009.01.002. Google Scholar

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