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The title says is all. This question is in the contest of an incompressible Navier-Stokes solver.

Specifically, I am currently working on a new solver while referring myself to an older code for guidance. The use of high-order methods - WENO5 for convection and 4th order finite difference other derivatives - is compulsory. My advisor wants me to use a staggered grid, which complicates the problem but more importantly, will add an additional ~15% computing for each time step (e.g. $\dfrac{\partial u}{\partial x}$ needs to be computed on the $u, v, w$ and pressure points instead of just one grid point for a regular grid).

I remember the first advantage of the staggered grid was to avoid checkerboarding oscillatory solutions and pressure-velocity coupling. But when using higher order methods, doesn't that emilinate the problem automatically? Also, I think I am willing to use staggered grids if they provide some real numerical advantage over collocated grids.

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  • $\begingroup$ My experience and knowledge is that to use collocated grids with any standard (lower or higher order) methods for this problem, you must modify (stabilize) your Navier-Stokes equation by e.g. introducing artificial compressibility and so on, it means with some additional effort. This needs not to be necessary when using staggered grids. $\endgroup$ – Peter Frolkovič Oct 26 '16 at 18:48
  • $\begingroup$ @PeterFrolkovič Thanks for the input. Would you have any references that covers this aspect? It'll be very helpfull $\endgroup$ – solalito Oct 27 '16 at 7:00
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Let me formulate my remark as an answer (and make it more precise):

My experience and knowledge is that to use collocated grids with any standard (lower or higher order) methods for this problem, you must modify (stabilize) your Navier-Stokes equation. it means with some additional effort. This needs not to be necessary when using staggered grids.

The references and some summary I am familiar with are:

S. Patankar: Numerical Heat Transfer and Fluid Flow. 1980 - see chapter 6 "Calculation of flow field" where the problem of checkerboard pattern of pressure values is mentioned for "collocated" grids (i.e. all unknowns at the same grid points) and where only single remedy is proposed - staggered grids, and no modifications of equations or discretization methods is necessary.

J.H. Ferziger, M. Peric: Computational Methods for Fluid Dynamics. 2002 (3rd edition) - it can be seen as a continuation of the book of Patankar. See Chapter 7 on Solution of the Navier-Stokes equation and sections 7.2 on Choice of Variable Arrangement on the Gird, and sections afterwards.

It is noted there that between 1960 and 1980 after introducing staggered grids very few works used the collocated grids. The collocated grids began to be again popular with "projection methods" that exist in many variants and that are well discussed in the book of Ferziger and Peric. If I simplify it the basic idea is to derive and use a Poisson equation to compute the pressure and to update afterwards the velocity field to be conservative.

An introduction of stabilizations (sometimes compared to artificial compressibility) on collocated grids is known to me from papers like:

Schneider, G.E., Raw, M.J.: Control volume finite-element method for heat transfer and fluid flow using collocated variables — 1. Computational procedure. Numer. Heat Transf. 11, 363–390 (1987)

Karimian, S.M., Schneider, G.E.: Pressure-based control-volume finite element method for flow at all speeds. AIAA J. 33(9), 1611– 1618 (1995)

Nägele, S., Wittum, G.: On the influence of different stabilisation methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 224, 100–116 (2007)

So my quite subjective summary might be that when using staggered grids, no additional modification to some "standard" numerical methods is required. Opposite to that for collocated grids some additional efforts is necessary.

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  • $\begingroup$ At least the books you cite (perhaps the papers too) deal with at most second order numerics, and thus do not answer the question fully in my opinion. $\endgroup$ – akid Oct 28 '16 at 12:42

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