# American Institute of Mathematical Sciences

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

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

 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.
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

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##### 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
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