Stability of approximate projection methods on cell-centered grids

Projection methods are a popular class of methods for solving the incompressible Navier-Stokes equations. If a cell-centered grid is chosen, in order to use high-resolution methods for the advection terms, performing the projection exactly is problematic. An attractive alternative is to use an approximate projection, in which the velocity is required to be only approximately discretely divergence-free. The stability of the cell-centered, approximate projection is highly sensitive to the method used to update the pressure and compute the pressure gradient. This is demonstrated by analyzing a model problem and conducting numerical simulations of the Navier-Stokes equations.

[1]  P. Colella,et al.  A Conservative Adaptive Projection Method for the Variable Density Incompressible Navier-Stokes Equations , 1998 .

[2]  M. Minion.  A Projection Method for Locally Refined Grids , 1996 .

[3]  Brian Wetton.  Analysis of the Spatial Error for a Class of Finite Difference Methods for Viscous Incompressible Flow , 1997 .

[4]  Jingyi Zhu,et al.  The Second-Order Projection Method for the Backward-Facing Step Flow , 1995 .

[5]  P. Colella.  Multidimensional upwind methods for hyperbolic conservation laws , 1990 .

[6]  Randall J. LeVeque,et al.  An Immersed Interface Method for Incompressible Navier-Stokes Equations , 2003, SIAM J. Sci. Comput..

[7]  P. Moin,et al.  Application of a Fractional-Step Method to Incompressible Navier-Stokes Equations , 1984 .

[8]  F. Harlow,et al.  Numerical Calculation of Time‐Dependent Viscous Incompressible Flow of Fluid with Free Surface , 1965 .

[9]  A. Chorin.  Numerical solution of the Navier-Stokes equations , 1968 .

[10]  John B. Bell,et al.  A Numerical Method for the Incompressible Navier-Stokes Equations Based on an Approximate Projection , 1996, SIAM J. Sci. Comput..

[11]  Alexandre Joel Chorin,et al.  On the Convergence of Discrete Approximations to the Navier-Stokes Equations* , 1989 .

[12]  Jie Shen.  On error estimates of projection methods for Navier-Stokes equations: first-order schemes , 1992 .

[13]  John B. Bell,et al.  An Adaptive Mesh Projection Method for Viscous Incompressible Flow , 1997, SIAM J. Sci. Comput..

[14]  P. Colella,et al.  A second-order projection method for the incompressible navier-stokes equations , 1989 .

[15]  John C. Strikwerda,et al.  The Accuracy of the Fractional Step Method , 1999, SIAM J. Numer. Anal..

[16]  R. LeVeque.  Wave Propagation Algorithms for Multidimensional Hyperbolic Systems , 1997 .

[17]  M. Minion,et al.  Accurate projection methods for the incompressible Navier—Stokes equations , 2001 .

[18]  Jian‐Guo Liu,et al.  Projection method I: convergence and numerical boundary layers , 1995 .

[19]  J. B. Perot,et al.  An analysis of the fractional step method , 1993 .

[20]  Zhilin Li,et al.  The immersed interface method for the Navier-Stokes equations with singular forces , 2001 .

[21]  Jie Shen,et al.  On error estimates of the projection methods for the Navier-Stokes equations: Second-order schemes , 1996, Math. Comput..

[22]  E Weinan,et al.  Projection Method II: Godunov--Ryabenki Analysis , 1996 .

[23]  J. Marsden,et al.  A mathematical introduction to fluid mechanics , 1979 .

[24]  Brian Wetton,et al.  Error Analysis for Chorin's Original Fully Discrete Projection Method and Regularizations in Space and Time , 1997 .

[25]  J. Kan.  A second-order accurate pressure correction scheme for viscous incompressible flow , 1986 .

[26]  S. Orszag,et al.  Boundary conditions for incompressible flows , 1986 .

[27]  John B. Bell,et al.  Approximate Projection Methods: Part I. Inviscid Analysis , 2000, SIAM J. Sci. Comput..