American Institute of Mathematical Sciences

Advanced
October  2014, 7(5): 993-1023. doi: 10.3934/dcdss.2014.7.993

Convergent finite differences for 1D viscous isentropic flow in Eulerian coordinates

Trygve K. Karper 1,

1. 

Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, 7491, Norway

Received  March 2013 Published  May 2014

We construct a new finite difference method for the flow of ideal viscous isentropic gas in one spatial dimension. For the continuity equation, the method is a standard upwind discretization. For the momentum equation, the method is an uncommon upwind discretization, where the moment and the velocity are solved on dual grids. Our main result is convergence of the method as discretization parameters go to zero. Convergence is proved by adapting the mathematical existence theory of Lions and Feireisl to the numerical setting.
Keywords: compensated compactness., compactness, isentropic, compressible Navier-Stokes, finite elements, weak convergence, convergence, upwind, Ideal gas, finite volume, discontinuous Galerkin.
Mathematics Subject Classification: Primary: 35Q30, 74S05; Secondary: 65M1.
Citation: Trygve K. Karper. Convergent finite differences for 1D viscous isentropic flow in Eulerian coordinates. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 993-1023. doi: 10.3934/dcdss.2014.7.993
References:
[1]

G.-Q. Chen, D. Hoff and K. Trivisa, Global solutions of the compressible Navier-Stokes equations with large discontinuous initial data,, Comm. Partial Differential Equations, 25 (2000), 2233.  doi: 10.1080/03605300008821583.  Google Scholar

[2]

R. Eymard, T. Gallouët, R. Herbin and J.-C. Latché, Convergence of the MAC scheme for the compressible Stokes equations,, SIAM J. Numer. Anal., 48 (2010), 2218.  doi: 10.1137/090779863.  Google Scholar

[3]

R. Eymard, T. Gallouët, R. Herbin and J.-C. Latché, A convergent finite element-finite volume scheme for the compressible Stokes problem. Part II: The isentropic case,, Math. Comp., 79 (2010), 649.  doi: 10.1090/S0025-5718-09-02310-2.  Google Scholar

[4]

E. Feireisl., Dynamics of Viscous Compressible Fluids,, Oxford Lecture Series in Mathematics and its Applications, (2004).   Google Scholar

[5]

T. Gallouët, R. Herbin and J.-C. Latché, A convergent finite element-finite volume scheme for the compressible Stokes problem. Part I: The isothermal case,, Math. Comp., 78 (2009), 1333.  doi: 10.1090/S0025-5718-09-02216-9.  Google Scholar

[6]

T. Gallouët, L. Gestaldo, R. Herbin and J.-C. Latché, An unconditionally stable pressure correction scheme for the compressible barotropic Navier-Stokes equations,, M2AN Math. Model. Numer. Anal., 42 (2008), 303.  doi: 10.1051/m2an:2008005.  Google Scholar

[7]

D. Hoff, Global existence for $1$D, compressible, isentropic Navier-Stokes equations with large initial data,, Trans. Amer. Math. Soc., 303 (1987), 169.  doi: 10.2307/2000785.  Google Scholar

[8]

H. K. Jenssen and T. K. Karper, One-dimensional compressible flow with temperature dependent transport coefficients,, SIAM J. Math. Anal., 42 (2010), 904.  doi: 10.1137/090763135.  Google Scholar

[9]

J. I. Kanel', A model system of equations for the one-dimensional motion of a gas,, Differencial' nye Uravnenija., 4 (1968), 721.   Google Scholar

[10]

K. Karlsen and T. K. Karper, A convergent nonconforming method for compressible flow,, SIAM J. Numer. Anal., 48 (2010), 1847.  doi: 10.1137/09076310X.  Google Scholar

[11]

K. Karlsen and T. K. Karper, Convergence of a mixed method for a semi-stationary compressible Stokes system,, Math. Comp., 80 (2011), 1459.  doi: 10.1090/S0025-5718-2010-02446-9.  Google Scholar

[12]

K. Karlsen and T. K. Karper, A convergent mixed method for the Stokes approximation of viscous compressible flow,, IMA J. Numer. Anal., 32 (2011), 725.  doi: 10.1093/imanum/drq048.  Google Scholar

[13]

T. K. Karper, A convergent FEM-DG method for the compressible Navier-Stokes equations,, Numer. Math., 125 (2013), 441.  doi: 10.1007/s00211-013-0543-7.  Google Scholar

[14]

A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas,, Prikl. Mat. Meh., 41 (1977), 282.   Google Scholar

[15]

P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2: Compressible Models,, Oxford University Press, (1998).   Google Scholar

[16]

A. Novotný and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow,, Oxford University Press, (2004).   Google Scholar

[17]

J. Serrin, Mathematical principles of classical fluid mechanics,, in 1959 Handbuch der Physik (herausgegeben von S. Flügge), (): 125.   Google Scholar

[18]

R. Zarnowski and D. Hoff, A finite-difference scheme for the Navier-Stokes equations of one-dimensional, isentropic, compressible flow,, SIAM J. Numer. Anal., 28 (1991), 78.  doi: 10.1137/0728004.  Google Scholar

[19]

J. Zhao and D. Hoff, A convergent finite-difference scheme for the Navier-Stokes equations of one-dimensional, nonisentropic, compressible flow,, SIAM J. Numer. Anal., 31 (1994), 1289.  doi: 10.1137/0731067.  Google Scholar

[20]

J. J. Zhao and D. Hoff, Convergence and error bound analysis of a finite-difference scheme for the one-dimensional Navier-Stokes equations,, in Nonlinear Evolutionary Partial Differential Equations (Beijing, (1993), 625.   Google Scholar

show all references

References:
[1]

G.-Q. Chen, D. Hoff and K. Trivisa, Global solutions of the compressible Navier-Stokes equations with large discontinuous initial data,, Comm. Partial Differential Equations, 25 (2000), 2233.  doi: 10.1080/03605300008821583.  Google Scholar

[2]

R. Eymard, T. Gallouët, R. Herbin and J.-C. Latché, Convergence of the MAC scheme for the compressible Stokes equations,, SIAM J. Numer. Anal., 48 (2010), 2218.  doi: 10.1137/090779863.  Google Scholar

[3]

R. Eymard, T. Gallouët, R. Herbin and J.-C. Latché, A convergent finite element-finite volume scheme for the compressible Stokes problem. Part II: The isentropic case,, Math. Comp., 79 (2010), 649.  doi: 10.1090/S0025-5718-09-02310-2.  Google Scholar

[4]

E. Feireisl., Dynamics of Viscous Compressible Fluids,, Oxford Lecture Series in Mathematics and its Applications, (2004).   Google Scholar

[5]

T. Gallouët, R. Herbin and J.-C. Latché, A convergent finite element-finite volume scheme for the compressible Stokes problem. Part I: The isothermal case,, Math. Comp., 78 (2009), 1333.  doi: 10.1090/S0025-5718-09-02216-9.  Google Scholar

[6]

T. Gallouët, L. Gestaldo, R. Herbin and J.-C. Latché, An unconditionally stable pressure correction scheme for the compressible barotropic Navier-Stokes equations,, M2AN Math. Model. Numer. Anal., 42 (2008), 303.  doi: 10.1051/m2an:2008005.  Google Scholar

[7]

D. Hoff, Global existence for $1$D, compressible, isentropic Navier-Stokes equations with large initial data,, Trans. Amer. Math. Soc., 303 (1987), 169.  doi: 10.2307/2000785.  Google Scholar

[8]

H. K. Jenssen and T. K. Karper, One-dimensional compressible flow with temperature dependent transport coefficients,, SIAM J. Math. Anal., 42 (2010), 904.  doi: 10.1137/090763135.  Google Scholar

[9]

J. I. Kanel', A model system of equations for the one-dimensional motion of a gas,, Differencial' nye Uravnenija., 4 (1968), 721.   Google Scholar

[10]

K. Karlsen and T. K. Karper, A convergent nonconforming method for compressible flow,, SIAM J. Numer. Anal., 48 (2010), 1847.  doi: 10.1137/09076310X.  Google Scholar

[11]

K. Karlsen and T. K. Karper, Convergence of a mixed method for a semi-stationary compressible Stokes system,, Math. Comp., 80 (2011), 1459.  doi: 10.1090/S0025-5718-2010-02446-9.  Google Scholar

[12]

K. Karlsen and T. K. Karper, A convergent mixed method for the Stokes approximation of viscous compressible flow,, IMA J. Numer. Anal., 32 (2011), 725.  doi: 10.1093/imanum/drq048.  Google Scholar

[13]

T. K. Karper, A convergent FEM-DG method for the compressible Navier-Stokes equations,, Numer. Math., 125 (2013), 441.  doi: 10.1007/s00211-013-0543-7.  Google Scholar

[14]

A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas,, Prikl. Mat. Meh., 41 (1977), 282.   Google Scholar

[15]

P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2: Compressible Models,, Oxford University Press, (1998).   Google Scholar

[16]

A. Novotný and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow,, Oxford University Press, (2004).   Google Scholar

[17]

J. Serrin, Mathematical principles of classical fluid mechanics,, in 1959 Handbuch der Physik (herausgegeben von S. Flügge), (): 125.   Google Scholar

[18]

R. Zarnowski and D. Hoff, A finite-difference scheme for the Navier-Stokes equations of one-dimensional, isentropic, compressible flow,, SIAM J. Numer. Anal., 28 (1991), 78.  doi: 10.1137/0728004.  Google Scholar

[19]

J. Zhao and D. Hoff, A convergent finite-difference scheme for the Navier-Stokes equations of one-dimensional, nonisentropic, compressible flow,, SIAM J. Numer. Anal., 31 (1994), 1289.  doi: 10.1137/0731067.  Google Scholar

[20]

J. J. Zhao and D. Hoff, Convergence and error bound analysis of a finite-difference scheme for the one-dimensional Navier-Stokes equations,, in Nonlinear Evolutionary Partial Differential Equations (Beijing, (1993), 625.   Google Scholar

[1]

Zhilei Liang. Convergence rate of solutions to the contact discontinuity for the compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1907-1926. doi: 10.3934/cpaa.2013.12.1907
[2]

Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020025
[3]

Boris Haspot, Ewelina Zatorska. From the highly compressible Navier-Stokes equations to the porous medium equation -- rate of convergence. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3107-3123. doi: 10.3934/dcds.2016.36.3107
[4]

Matúš Tibenský, Angela Handlovičová. Convergence analysis of the discrete duality finite volume scheme for the regularised Heston model. Discrete & Continuous Dynamical Systems - S, 2019  doi: 10.3934/dcdss.2020226
[5]

Boris Andreianov, Mostafa Bendahmane, Kenneth H. Karlsen, Charles Pierre. Convergence of discrete duality finite volume schemes for the cardiac bidomain model. Networks & Heterogeneous Media, 2011, 6 (2) : 195-240. doi: 10.3934/nhm.2011.6.195
[6]

Matthew Paddick. The strong inviscid limit of the isentropic compressible Navier-Stokes equations with Navier boundary conditions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2673-2709. doi: 10.3934/dcds.2016.36.2673
[7]

Roberta Bianchini, Roberto Natalini. Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations. Kinetic & Related Models, 2019, 12 (1) : 133-158. doi: 10.3934/krm.2019006
[8]

Caterina Calgaro, Meriem Ezzoug, Ezzeddine Zahrouni. Stability and convergence of an hybrid finite volume-finite element method for a multiphasic incompressible fluid model. Communications on Pure & Applied Analysis, 2018, 17 (2) : 429-448. doi: 10.3934/cpaa.2018024
[9]

Grzegorz Karch, Maria E. Schonbek, Tomas P. Schonbek. Singularities of certain finite energy solutions to the Navier-Stokes system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 189-206. doi: 10.3934/dcds.2020008
[10]

Pavel I. Plotnikov, Jan Sokolowski. Optimal shape control of airfoil in compressible gas flow governed by Navier-Stokes equations. Evolution Equations & Control Theory, 2013, 2 (3) : 495-516. doi: 10.3934/eect.2013.2.495
[11]

Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602
[12]

Tadahisa Funaki, Yueyuan Gao, Danielle Hilhorst. Convergence of a finite volume scheme for a stochastic conservation law involving a $Q$-brownian motion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1459-1502. doi: 10.3934/dcdsb.2018159
[13]

Mostafa Bendahmane, Mauricio Sepúlveda. Convergence of a finite volume scheme for nonlocal reaction-diffusion systems modelling an epidemic disease. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 823-853. doi: 10.3934/dcdsb.2009.11.823
[14]

Rajesh Kumar, Jitendra Kumar, Gerald Warnecke. Convergence analysis of a finite volume scheme for solving non-linear aggregation-breakage population balance equations. Kinetic & Related Models, 2014, 7 (4) : 713-737. doi: 10.3934/krm.2014.7.713
[15]

Yanmin Mu. Convergence of the compressible isentropic magnetohydrodynamic equations to the incompressible magnetohydrodynamic equations in critical spaces. Kinetic & Related Models, 2014, 7 (4) : 739-753. doi: 10.3934/krm.2014.7.739
[16]

Mohamed Alahyane, Abdelilah Hakim, Amine Laghrib, Said Raghay. Fluid image registration using a finite volume scheme of the incompressible Navier Stokes equation. Inverse Problems & Imaging, 2018, 12 (5) : 1055-1081. doi: 10.3934/ipi.2018044
[17]

Haibo Cui, Zhensheng Gao, Haiyan Yin, Peixing Zhang. Stationary waves to the two-fluid non-isentropic Navier-Stokes-Poisson system in a half line: Existence, stability and convergence rate. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4839-4870. doi: 10.3934/dcds.2016009
[18]

Tianliang Hou, Yanping Chen. Superconvergence for elliptic optimal control problems discretized by RT1 mixed finite elements and linear discontinuous elements. Journal of Industrial & Management Optimization, 2013, 9 (3) : 631-642. doi: 10.3934/jimo.2013.9.631
[19]

Zhenhua Guo, Zilai Li. Global existence of weak solution to the free boundary problem for compressible Navier-Stokes. Kinetic & Related Models, 2016, 9 (1) : 75-103. doi: 10.3934/krm.2016.9.75
[20]

Jishan Fan, Fucai Li, Gen Nakamura. Convergence of the full compressible Navier-Stokes-Maxwell system to the incompressible magnetohydrodynamic equations in a bounded domain. Kinetic & Related Models, 2016, 9 (3) : 443-453. doi: 10.3934/krm.2016002

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (21)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences