July & August  2021, 20(7&8): 2751-2763. doi: 10.3934/cpaa.2021084

On the invariant region for compressible Euler equations with a general equation of state

1. 

Iowa State University, Mathematics Department, Ames, IA 50011, USA

2. 

Otto-von-Guericke-Universität, Universitätsplatz 2, Magdeburg, 39106, Germany

* Corresponding author

Dedicated to Professor Shuxing Chen on the occasion of his 80th birthday

Received  February 2021 Revised  April 2021 Published  July & August 2021 Early access  May 2021

Fund Project: Hailiang Liu was partially supported by the National Science Foundation under Grant DMS1812666

The state space for solutions of the compressible Euler equations with a general equation of state is examined. An arbitrary equation of state is allowed, subject only to the physical requirements of thermodynamics. An invariant region of the resulting Euler system is identified and the convexity property of this region is justified by using only very minimal thermodynamical assumptions. Finally, we show how an invariant-region-preserving (IRP) limiter can be constructed for use in high order finite-volume type schemes to solve the compressible Euler equations with a general constitutive relation.

Citation: Hailiang Liu, Ferdinand Thein. On the invariant region for compressible Euler equations with a general equation of state. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2751-2763. doi: 10.3934/cpaa.2021084
References:
[1]

J. H. Dymond and R. Malhotra, The Tait equation: 100 years on, Int. J. Thermophys., 9 (1988), 941-951. 

[2]

L. C. Evans, Entropy and Partial Differential Equations, Lecture notes, 2010.

[3]

H. Frid, Invariant regions under Lax-Friedrichs scheme for multidimensional systems of conservation laws, Discrete Contin. Dyn. Syst., 1 (1995), 585-593.  doi: 10.3934/dcds.1995.1.585.

[4]

H. Frid, Maps of Convex Sets and Invariant Regions for Finite-Difference Systems of Conservation Laws, Arch. Ration. Mech. Anal., 160 (2001), 245-269.  doi: 10.1007/s002050100166.

[5]

J. L. Guermond and B. Popov, Invariant domains and first-order continuous finite element approximation for hyperbolic systems, SIAM J. Numer. Anal., 54 (2016), 2466-2489.  doi: 10.1137/16M1074291.

[6]

M. Hantke and F. Thein, Why condensation by compression in pure water vapor cannot occur in an approach based on Euler equations, Quart. Appl. Math., 73 (2015), 575-591.  doi: 10.1090/qam/1393.

[7]

D. Hoff, Invariant regions for systems of conservation laws, Trans. Amer. Math. Soc., 289 (1985), 591-610.  doi: 10.2307/2000254.

[8]

M. J. IvingsD. M. Causon and E. F. Toro, On Riemann solvers for compressible liquids, Int. J. Numer. Methods Fluids, 28 (1998), 395-418.  doi: 10.1002/(SICI)1097-0363(19980915)28:3<395::AID-FLD718>3.0.C0;2-S.

[9]

Y. Jiang and H. Liu, An Invariant-region-preserving (IRP) Limiter to DG Methods for Compressible Euler Equations, Springer, Cham, 2018. doi: 10.1007/978-3-319-91548-7.

[10]

Y. Jiang and H. Liu, Invariant-region-preserving DG methods for multi-dimensional hyperbolic conservation law systems, with an application to compressible Euler equations, J. Comput. Phys., 373 (2018), 385-409.  doi: 10.1016/J.JCP.2018.03.004.

[11]

Y. Jiang and H. Liu, An invariant region preserving limiter for DG schemes to isentropic Euler equations, Numer. method. PDEs, 35 (2019), 5-33.  doi: 10.1002/num.22274.

[12]

B. Khobalatte and B. Perthame, Maximum principle on the entropy and second-order kinetic schemes, Math. Comput., 62 (1994), 119-131.  doi: 10.2307/2153399.

[13]

L. D. Landau and E. M. Lifšic, Lehrbuch der Theoretischen Physik. Band V: Statistische Physik, Akademie-Verlag, Berlin, 1987.

[14]

R. Menikoff and B. J. Plohr, The Riemann problem for fluid flow of real materials, Rev. Mod. Phys., 61 (1989), 75-130.  doi: 10.1103/RevModPhys.61.75.

[15]

S. Müller and A. Voss, The Riemann Problem for the Euler Equations with Nonconvex and Nonsmooth Equation of State: Construction of Wave Curves, SIAM J. Sci. Comput., 28 (2006), 651-681.  doi: 10.1137/040619909.

[16]

B. Perthame and C. W. Shu, On positivity preserving finite volume schemes for Euler equations, Numer. Math., 73 (1996), 119-130.  doi: 10.1007/s002110050187.

[17]

R. SaurelP. Cocchi and P. Butler, Numerical Study of Cavitation in the Wake of a Hypervelocity Underwater Projectile, J. Propul. Power, 15 (1999), 513-522. 

[18]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, New York-Berlin, 1983.

[19]

F. Thein, Results for Two Phase Flows with Phase Transition, Dissertation, Otto-von-Guericke University of Magdeburg, 2018.

[20]

E. Tadmor, A minimum entropy principle in the gas dynamics equations, Appl. Numer. Math., 2 (1986), 211-219.  doi: 10.1016/0168-9274(86)90029-2.

[21]

X. Zhang and C. W. Shu, On maximum-principle-satisfying high order schemes for scalar conservation laws, J. Comput. Phys., 229 (2010), 3091-3120.  doi: 10.1016/j.jcp.2009.12.030.

[22]

X. Zhang and C. W. Shu, On positivity preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes, J. Comput. Phys., 229 (2010), 8918-8934.  doi: 10.1016/j.jcp.2010.08.016.

[23]

X. Zhang and C. W. Shu, A minimum entropy principle of high order schemes for gas dynamics equations, Numer. Math., 121 (2012), 545-563.  doi: 10.1007/s00211-011-0443-7.

[24]

X. ZhangY. Xia and C. W. Shu, Maximum-principle-satisfying and positivity-preserving high order discontinuous Galerkin schemes for conservation laws on triangular meshes, J. Sci. Comput., 50 (2012), 29-62.  doi: 10.1007/s10915-011-9472-8.

show all references

References:
[1]

J. H. Dymond and R. Malhotra, The Tait equation: 100 years on, Int. J. Thermophys., 9 (1988), 941-951. 

[2]

L. C. Evans, Entropy and Partial Differential Equations, Lecture notes, 2010.

[3]

H. Frid, Invariant regions under Lax-Friedrichs scheme for multidimensional systems of conservation laws, Discrete Contin. Dyn. Syst., 1 (1995), 585-593.  doi: 10.3934/dcds.1995.1.585.

[4]

H. Frid, Maps of Convex Sets and Invariant Regions for Finite-Difference Systems of Conservation Laws, Arch. Ration. Mech. Anal., 160 (2001), 245-269.  doi: 10.1007/s002050100166.

[5]

J. L. Guermond and B. Popov, Invariant domains and first-order continuous finite element approximation for hyperbolic systems, SIAM J. Numer. Anal., 54 (2016), 2466-2489.  doi: 10.1137/16M1074291.

[6]

M. Hantke and F. Thein, Why condensation by compression in pure water vapor cannot occur in an approach based on Euler equations, Quart. Appl. Math., 73 (2015), 575-591.  doi: 10.1090/qam/1393.

[7]

D. Hoff, Invariant regions for systems of conservation laws, Trans. Amer. Math. Soc., 289 (1985), 591-610.  doi: 10.2307/2000254.

[8]

M. J. IvingsD. M. Causon and E. F. Toro, On Riemann solvers for compressible liquids, Int. J. Numer. Methods Fluids, 28 (1998), 395-418.  doi: 10.1002/(SICI)1097-0363(19980915)28:3<395::AID-FLD718>3.0.C0;2-S.

[9]

Y. Jiang and H. Liu, An Invariant-region-preserving (IRP) Limiter to DG Methods for Compressible Euler Equations, Springer, Cham, 2018. doi: 10.1007/978-3-319-91548-7.

[10]

Y. Jiang and H. Liu, Invariant-region-preserving DG methods for multi-dimensional hyperbolic conservation law systems, with an application to compressible Euler equations, J. Comput. Phys., 373 (2018), 385-409.  doi: 10.1016/J.JCP.2018.03.004.

[11]

Y. Jiang and H. Liu, An invariant region preserving limiter for DG schemes to isentropic Euler equations, Numer. method. PDEs, 35 (2019), 5-33.  doi: 10.1002/num.22274.

[12]

B. Khobalatte and B. Perthame, Maximum principle on the entropy and second-order kinetic schemes, Math. Comput., 62 (1994), 119-131.  doi: 10.2307/2153399.

[13]

L. D. Landau and E. M. Lifšic, Lehrbuch der Theoretischen Physik. Band V: Statistische Physik, Akademie-Verlag, Berlin, 1987.

[14]

R. Menikoff and B. J. Plohr, The Riemann problem for fluid flow of real materials, Rev. Mod. Phys., 61 (1989), 75-130.  doi: 10.1103/RevModPhys.61.75.

[15]

S. Müller and A. Voss, The Riemann Problem for the Euler Equations with Nonconvex and Nonsmooth Equation of State: Construction of Wave Curves, SIAM J. Sci. Comput., 28 (2006), 651-681.  doi: 10.1137/040619909.

[16]

B. Perthame and C. W. Shu, On positivity preserving finite volume schemes for Euler equations, Numer. Math., 73 (1996), 119-130.  doi: 10.1007/s002110050187.

[17]

R. SaurelP. Cocchi and P. Butler, Numerical Study of Cavitation in the Wake of a Hypervelocity Underwater Projectile, J. Propul. Power, 15 (1999), 513-522. 

[18]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, New York-Berlin, 1983.

[19]

F. Thein, Results for Two Phase Flows with Phase Transition, Dissertation, Otto-von-Guericke University of Magdeburg, 2018.

[20]

E. Tadmor, A minimum entropy principle in the gas dynamics equations, Appl. Numer. Math., 2 (1986), 211-219.  doi: 10.1016/0168-9274(86)90029-2.

[21]

X. Zhang and C. W. Shu, On maximum-principle-satisfying high order schemes for scalar conservation laws, J. Comput. Phys., 229 (2010), 3091-3120.  doi: 10.1016/j.jcp.2009.12.030.

[22]

X. Zhang and C. W. Shu, On positivity preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes, J. Comput. Phys., 229 (2010), 8918-8934.  doi: 10.1016/j.jcp.2010.08.016.

[23]

X. Zhang and C. W. Shu, A minimum entropy principle of high order schemes for gas dynamics equations, Numer. Math., 121 (2012), 545-563.  doi: 10.1007/s00211-011-0443-7.

[24]

X. ZhangY. Xia and C. W. Shu, Maximum-principle-satisfying and positivity-preserving high order discontinuous Galerkin schemes for conservation laws on triangular meshes, J. Sci. Comput., 50 (2012), 29-62.  doi: 10.1007/s10915-011-9472-8.

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