Collocated Mesh — CAE Glossary

A collocated mesh (Collocated Mesh) is a method in which all physical quantities such as velocity $u, v, w$, pressure $p$, and temperature $T$ are stored at the same location at the cell center. "Co-located" means "placed in the same location."

Good question. Historically, it was not like that. During the heyday of structured grid CFD in the 1960s–80s, the staggered mesh (Staggered Mesh) was the dominant approach. In staggered grids, velocity components are placed at cell faces and pressure is placed at the cell center. By intentionally offsetting the variable locations, you could obtain a numerically stable pressure field.

Staggered grids certainly suppress pressure oscillations naturally on structured grids. But it has a major weakness: it is extremely difficult to apply to unstructured grids (triangles, tetrahedra, polyhedra, etc.). When you want to mesh complex shapes like automobile bodies or turbine blades into a mesh, unstructured grids are essential, right? That's where collocated grids come in.

Here's a summary:

This is the famous Checkerboard Pressure Problem. Let's think about it in one dimension. When we discretize the pressure gradient term in the momentum equation using central differences, the pressure gradient at cell $i$ becomes

Exactly! For example, a pressure distribution of $p = [100, 0, 100, 0, 100, \ldots]$ would look like zero gradient in central differences. The pressure correction using the continuity equation cannot eliminate this oscillation mode either. As a result, the solution ends up with a physically impossible checkerboard pressure field. In 2D it looks just like a chessboard, so it's called "checkerboard."

In staggered grids, velocity is located at cell faces. The pressure gradient at face $i+1/2$ for the velocity becomes

The Rhie-Chow Interpolation (Rhie-Chow Interpolation), proposed by Rhie and Chow in 1983, is the solution. It is used in combination with the SIMPLE-family pressure-correction algorithm. The basic idea is this:

Roughly speaking, we're artificially recreating the pressure gradient that would have been obtained on a staggered grid on a collocated grid. The pressure gradient at the face $(\partial p/\partial x)_f$ is calculated from the pressure difference between adjacent cells, so it directly sees $p_i$ and $p_{i+1}$. On the other hand, the gradient at the cell center interpolated to the face $\overline{(\partial p/\partial x)}_f$ only has information from alternating cells. By taking the difference of these two, a damping (attenuation) effect for the checkerboard mode is created.

That's a sharp observation. The Rhie-Chow correction term depends on grid spacing $\Delta x$, so as the grid is refined, the correction amount becomes smaller and its impact on accuracy disappears. In other words, grid convergence is preserved. However, in unsteady calculations, there is a known issue of time-step dependence, with excessive damping as $\Delta t \to 0$. This was corrected in improved versions by Choi (1999) and others, but in practice it rarely becomes a major issue with default settings.

There are several important points:

Right, all current major solvers assume collocated grids, so you rarely see a user-facing option to "choose between collocated and staggered." Rhie-Chow interpolation is automatically applied internally by the solver. However, it's important to recognize when your solution shows pressure oscillation-like behavior—that's when you might think "this could be checkerboard." It becomes a cue to improve mesh quality or review discretization schemes.

Perfect summary. What you should remember is that the mesh arrangement method directly affects discretization accuracy and stability. Knowing "why this grid arrangement?" and "what artificial processing is happening behind the scenes?" dramatically improves your ability to interpret results and troubleshoot.