Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

The use of Richardson extrapolation for the numerical solution of low Mach number flow in confined regions

  • 71 Accesses

  • 2 Citations


We use artificial compressibility together with Richardson extrapolation in the Mach numberM as a method for solving the time dependent Navier-Stokes equation for very low Mach number flow and for incompressible flow. The question of what boundary conditions one should use for low Mach number flow, especially at inflow and outflow boundaries, is investigated theoretically, and boundary layer suppressing boundary conditions are derived. For the case of linearization around a constant flow we show that the low Mach number solution will converge with the rateO(M2) to the true incompressible solution, provided that we choose the boundary conditions correctly. The results of numerical calculations for the time dependent, nonlinear equations and for flow situations with time dependent inflow velocity profiles are presented. The convergence rateM 2 to incompressible solution is numerically confirmed. It is also shown that using Richardson extrapolation toM 2= 0 in order to derive a solution with very small divergence can with good result be carried through withM 2 as large as 0.1 and 0.05. As the time step in numerical methods must be chosen approximately such thatΔt · (i/(M Δx)−v/Δx 2) is in the stability region of the time stepping method, and asM 2=0.05 is sufficiently small to yield good results, the restriction on the time step due to the Mach number is not serious. Therefore the equations can be integrated very fast by explicit time stepping methods. This method for solving very low Mach number flow and incompressible flow is well suited to parallel processing.

This is a preview of subscription content, log in to check access.


  1. Chorin, A. J. (1967), A numerical method for solving incompressible viscous flow problems,J. Comput. Phys. 2, 12–26.

  2. Dettman, J. W. (1984).Applied Complex Variables, Dover Publications, Inc.

  3. Gustafsson, B., Kreiss, H.-O., and Oliger, J. (1990). The approximate solution of time-dependent problems (unpublished manuscript).

  4. Johansson, C. (1991a). Initial-boundary value problems for incompressible fluid flow, Doctoral Thesis in Numerical Analysis, Uppsala University, Department of Scientific Computing, Box 120, S-75104 Uppsala, Sweden.

  5. Johansson, C. (1991b). Boundary conditions for open boundaries for the incompressible Navier-Stokes equation, submitted toJ. Comput. Phys., (now published).

  6. Johansson, C. (1991c). Well-posedness in the generalized sense for the incompressible Navier-Stokes equation,J. Sci. Comput. 6(2), 101–127.

  7. Johansson, C. (1991d). Well-posedness in the generalized sense for boundary layer suppressing boundary conditions,J. Sci. Comput. 6(4), 391–114.

  8. Johansson, C. (1992). The numerical solution of low Mach number flow in confined regions by Richardson extrapolation, TRITA-NA-9207. Department of Numerical Analysis and Computing Science, Royal Institute of Technology, NADA, S-100 44 Stockholm, Sweden.

  9. Kreiss, H.-O., and Lorenz, J. (1989).Initial-Boundary Value Problems and the Navier-Stokes Equations, Academic Press.

  10. Kreiss, H.-O., Lorenz, J., and Naughton, M. J. (1991). Convergence of the solutions of the compressible to the solutions of the incompressible Navier-Stokes equations,Adv. Appl. Math. 12, 187–214.

  11. Naughton, M. J. (1986). On numerical boundary conditions for the Navier-Stokes equations, Ph.D. thesis, California Institute of Technology, Pasadena, California.

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Christer, B., Johansson, V. The use of Richardson extrapolation for the numerical solution of low Mach number flow in confined regions. J Sci Comput 8, 307–340 (1993).

Download citation

Key words

  • Navier-Stokes equations
  • incompressible
  • low Mach number
  • Richardson extrapolation
  • boundary layer
  • boundary condition
  • boundary layer suppressing boundary condition
  • inflow
  • outflow
  • open boundary