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. [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. [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. [4] E. Feireisl., Dynamics of Viscous Compressible Fluids,, Oxford Lecture Series in Mathematics and its Applications, (2004). [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. [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. [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. [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. [9] J. I. Kanel', A model system of equations for the one-dimensional motion of a gas,, Differencial' nye Uravnenija., 4 (1968), 721. [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. [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. [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. [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. [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. [15] P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2: Compressible Models,, Oxford University Press, (1998). [16] A. Novotný and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow,, Oxford University Press, (2004). [17] J. Serrin, Mathematical principles of classical fluid mechanics,, in 1959 Handbuch der Physik (herausgegeben von S. Flügge), (): 125. [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. [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. [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.

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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. [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. [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. [4] E. Feireisl., Dynamics of Viscous Compressible Fluids,, Oxford Lecture Series in Mathematics and its Applications, (2004). [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. [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. [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. [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. [9] J. I. Kanel', A model system of equations for the one-dimensional motion of a gas,, Differencial' nye Uravnenija., 4 (1968), 721. [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. [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. [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. [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. [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. [15] P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2: Compressible Models,, Oxford University Press, (1998). [16] A. Novotný and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow,, Oxford University Press, (2004). [17] J. Serrin, Mathematical principles of classical fluid mechanics,, in 1959 Handbuch der Physik (herausgegeben von S. Flügge), (): 125. [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. [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. [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.
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