# A staggered grid for an eigenvalue problem (linear stability analysis)

I'm interested in extending the concept of a staggered grid (commonly used to solve the incompressible Navier-Stokes equations) to a linear stability analysis context. For example, we can consider flow within a two-dimensional rectangular duct, as in Theofilis, Duck, and Owen in the Journal of Fluid Mechanics, 2004. The base flow here is $$\bar{u}_z(x,y)$$ only, just to simplify our discussion. The staggered grid (below) can be called a marker-and-cell (MAC) or a particle-in-cell (PIC) mesh. The inviscid temporal linear stability governing equations (solved as an eigenvalue problem) would be:

$$-i\beta\hat{u}-\partial\hat{p}/\partial x=-i\omega\hat{u}$$

$$-i\beta\hat{v}-\partial\hat{p}/\partial y=-i\omega\hat{v}$$

$$-\frac{\partial \bar{u}_z}{\partial x}\hat{u} - \frac{\partial\bar{u}_z}{\partial y}\hat{v}-i\beta\hat{w}-i\beta\hat{p}=-i\omega\hat{w}$$

$$\frac{\partial\hat{u}}{\partial x }+ \frac{\partial\hat{v}}{\partial y} - i\beta\hat{w}=0$$

Wouldn't combining differentiation with interpolation into the same matrix cause errors? Particularly around the boundaries!

A scheme could be:

1. Differentiation performed by finite differences
2. Discretize the continuity and z-momentum equations at the cell centers
3. Discretize the x-momentum equation at the x-edges
4. Discretize the y-momentum equation at the y-edges
5. Interpolate the $$u_x$$ values onto cells in the x direction and interpolate the $$u_y$$ values onto cells in the y direction for the z-momentum equation. Then multiply by the matrix for $$\partial\bar{u}_z/\partial x$$ or $$\partial\bar{u}_z/\partial y$$, respectively.
6. All other terms that need to be transferred from one grid to the other involve derivation and interpolation. For example, $$dp/dx$$ in the x-momentum equation could be treated as: create an interpolation matrix I from the pressure cells to the x-edges and then apply a differentiation matrix D to the result i.e., DI.

Steps 4 and 5 seem likely to introduce errors. Also, this scheme seems to be the only way to obtain a square eigenproblem.

Am I missing something?

Also, would the scheme need to be changed if a spatial linear stability analysis were performed instead?

I have seen Interpolation of velocities on staggered grid (in PIC), which explores interpolation, and is where I obtained the photo of the staggered grid.

• It is pretty standard to use staggered grids for eigenmode analysis, see, e.g., Baver, Myra, and Umansky, Comp. Phys. Comm. 182, 1610 (2011). BC for density-like variables are set at cell centers, BC for velocity-like variables are set at cell faces. Mar 2 at 1:18