A projection-difference scheme for the unsteady motion of a viscous barotropic gas

Authors

  • K.A. Zhukov
  • A.V. Popov

Keywords:

projection-difference scheme
finite element method
viscous gas
implicit difference schemes
equations of gas dynamics
unsteady flows

Abstract

A new implicit projection-difference scheme for the unsteady motion of a viscous barotropic gas is proposed in terms of Eulerian coordinates for the cases of one, two, and three spatial variables. The peculiarity of this scheme consists in the fact that we approximate the continuity equation written in terms of the logarithm of density, which allows one to ensure the positiveness of the density function for any parameters of the scheme. This scheme is two-layered. At each time step, a numerical solution is found by solving a linear system. The existence and uniqueness theorem for a numerical solution is proved without no additional conditions imposed on the time and spatial discretization parameters. It is shown that the scheme can be used in the problems with nonsmooth initial conditions in the one-dimensional case and in the cavity problem in the two-dimensional case. The importance of including the artificial viscosity in the approximation of the continuity equation is shown experimentally.

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Published

2014-10-22

Issue

Vol. 15 (2014): Issue 4.

Section

Section 1. Numerical methods and applications

Author Biographies

K.A. Zhukov

Lomonosov Moscow State University
• Researcher

A.V. Popov

Lomonosov Moscow State University
• Associate Professor

References

  1. Антонцев С.Н., Кажихов А.В., Монахов В.Н. Краевые задачи механики неоднородных жидкостей. Новосибирск: Наука, 1983.
  2. Вайгант В.А., Кажихов А.В. О существовании глобальных решений двумерных уравнений Навье-Стокса сжимаемой вязкой жидкости // Сиб. матем. журн. 1995. 36, № 6. 1283-1316.
  3. Жуков К.А, Попов А.В. Разностные и проекционно-разностные схемы для нестационарного движения вязкого слабосжимаемого газа // Вычислительные методы и программирование. 2012. 13. 67-73.
  4. Попов А.В., Жуков К.А. Неявная разностная схема для нестационарного движения вязкого баротропного газа // Вычислительные методы и программирование. 2013. 14. 516-523.
  5. Chambarel A., Bolvin H. Simulation of a compressible flow by the finite element method using a general parallel computing approach // Lecture Notes in Computer Science. Vol. 2329. Heidelberg: Springer, 2002. 920-929.
  6. Chung T.J. Computational fluid dynamics. New York: Cambridge Univ. Press, 2002.
  7. Donea J., Huerta A. Finite element methods for flow problems. Chichester: Wiley, 2003.
  8. Feireisl E., Novotn’y A., Petzeltova H. On the existence of globally defined weak solutions to the Navier-Stokes equations // J. Math. Fluid Mech. 2001. 3, N 4. 358-392.
  9. Hoff D. Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data // Arch. Rational Mech. Anal. 1995. 132, N 1. 1-14.
  10. Holton J.R., Hakim G.J. An introduction to dynamic meteorology. New York: Academic Press, 2013.
  11. Kellogg R.B., Liu B. A finite element method for the compressible Stokes equations // SIAM J. Num. Anal. 1996. 33, N 2. 780-789.
  12. Liu B. The analysis of a finite element method with streamline diffusion for the compressible Navier-Stokes equations // SIAM J. Num. Anal. 2000. 38, N 1. 1-16.
  13. Kirk B.S., Carey G.F. Development and validation of a SUPG finite element scheme for the compressible Navier-Stokes equations using a modified inviscid flux discretization // Int. J. Numer. Meth. Fluids. 2008. 57, N 3. 265-293.
  14. Kotteda V.M. K., Mittal S. Stabilized finite-element computation of compressible flow with linear and quadratic interpolation functions // Int. J. Numer. Meth. Fluids. 2014. 75, N 4. 273-294.
  15. Kweon J.R. Optimal error estimate for a mixed finite element method for compressible Navier-Stokes system // Appl. Numer. Math. 2003. 45, N 2. 275-292.
  16. Kellogg B., Liu B. The analysis of a finite element method for the Navier-Stokes equations with compressibility // Numerische Mathematik. 2000. 87, N 1. 153-170.
  17. Liu B. On a finite element method for unsteady compressible viscous flows // Numerical Methods for Partial Differential Equations. 2003. 19, N 2. 152-166.
  18. Liu B. On a finite element method for three-dimensional unsteady compressible viscous flows // Numerical Methods for Partial Differential Equations. 2004. 20, N 3. 432-449.
  19. Lions P.-L. Mathematical topics in fluid dynamics. Vol. 2. Compressible models. Oxford: Oxford Science Publication, 1998.
  20. Nazarov M., Hoffman J. Residual-based artificial viscosity for simulation of turbulent compressible flow using adaptive finite element methods // Int. J. Numer. Meth. Fluids. 2013. 71, N 3. 339-357.
  21. Matsumura A., Nishida T. Initial boundary value problems for the equations of motion of general fluids // Computing Methods in Applied Sciences and Engineering. Vol. 5. Amsterdam: North-Holland, 1982. 389-406.
  22. Matsumura A., Nishida T. Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids // Communications in Mathematical Physics. 1983. 89. 445-464.
  23. Salby M.L. Physics of the atmosphere and climate. New York: Cambridge Univ. Press, 2012.
  24. Valli A. Periodic and stationary solutions for compressible Navier-Stokes equations via a stability method // Ann. Scuola Norm. Sup. Pisa. 1983. 10, N 4. 607-647.
  25. Valli A. Mathematical results for compressible flows // Mathematical Topics in Fluid Mechanics. Pitman Research Notes in Mathematics. Ser. 274. Harlow: Longman, 1993. 193-229.
  26. Vynnycky M., Sharma A.K., Birgersson E. A finite-element method for the weakly compressible parabolized steady 3D Navier-Stokes equations in a channel with a permeable wall // Computers and Fluids. 2013. 81. 152-161.
  27. Zienkiewicz O.C., Taylor R.L. The finite element method. Vol. 3. Fluid dynamics. Oxford: Butterworth-Heinemann, 2000.