Non uniform coordinate spacing

[kurtsansom]

What is the name of the non-uniform grid spacing being used called? and where does definition come from?

I wrote out the equation defined in mesh.hpp and can kinda see whats going on, but am unsure how it was derived?

I have seen definitions for logarithmic spacing, geometric spacing, and I guess there could be extensions to lagrange polynomials, or any other mapping, but I am not sure how that affects the solution given what is mentioned in the wiki.

the wiki says:

While Athena++ uses formulae consistent with nonuniform mesh spacing, too large a mesh ratio should not be used; a non-fatal warning is raised if the value is not between 0.9 and 1.1 everywhere. The reconstruction algorithm reduces to first-order if the grid spacing changes too much from cell-to-cell, and moreover, the nonuniform truncation error can produce anomalous reflections of waves.

Note, the geometric spacing is equivalent to logarithmic spacing (uniform dx' in logspace given by the coordinate transformation x' = logb(x1) for some b>0 , b ≠ 1) only for a particular choice of (x1rat, x1min, and x1max).

Examples I would be interested in understanding would be clustering at the edges or clustering at the interior. What complications should I expect when trying to generalize the nonuniform mesh spacing?

[c-white]

The grid is always perfectly geometric in cell widths (i.e. differences in adjacent face-centered coordinates).

Let's say there are n cells with a ratio of r. Call the faces x₀, ..., x_n and the cell widths Δ₀, ..., Δ_n-1. Then Δ_i = Δ₀ rⁱ and so x_i = x₀ + Δ₀ ∑_j=0^i-1 r^j. We also have the boundary conditions x₀ = x_min and x_n = x_max.

To see the connection to logarithmic spacing, suppose the faces were uniformly log-spaced: ℓ(xⁱ) = ℓx_min + i ℓΔ for ℓΔ = (ℓx_max - ℓx_min) / n. Then xⁱ = x_min Δⁱ, and we have xⁱ⁺¹ / xⁱ is a constant, Δ. In this case, though, we also have Δ_i = x_i+1 - x_i = x_min Δⁱ (Δ - 1), and so Δ_i+1 / Δ_i is the constant Δ. That is, this is also geometric in cell widths with the association Δ₀ = x_min (Δ - 1) and r = Δ. Thus to get logarithmically spaced faces for given x_min, x_max, and n, choose r = (x_max / x_min)^1/n.

If you choose an r less than this value (i.e., closer to 1), the grid will be more uniformly spaced (less centrally concentrated). Conversely, larger r results in a more centrally concentrated grid. You could also choose r < 1 to make the grid concentrated toward x_max, though I've never seen anyone do this.

All that said, the x1rat parameter is there for convenience. Various parts of the code are often optimized for a uniform grid, and reconstruction might be able to do something slightly more accurate in that case, but any grid, even one that is not geometric in cell widths, will work.

[kurtsansom]

I found another implementation gridder that gives some equations for the geometric spacing. I tested out the function DefaultMeshGeneratorX1 and the gridder version in python to show they were equivalent to double precision. Then derived an equivalent expression to the one given in DefaultMeshGeneratorX1 using the gridder definitions and proved to myself that they were the same. I would provide some write up for the equations if anyone is interested.

@c-white thank you that does provide some insight, so logarithmic spacing is geometrically spaced given the extra definition for r

Tangent How do you get equations to show?, quick google search didn't help.

[c-white]

I cobbled together the equations in raw html, mostly using <var>, <sup>, and <sub> constructs. It's tedious to say the least, but there aren't many good alternatives. Many years ago github supported mathjax, but abandoned it out of laziness and/or foolishness. Given their responsiveness to users of late, I doubt we'll ever see equation rendering support.

[felker]

@kurtsansom would be great if you could write it up and we could add it to the Wiki. I remember debating the geometric vs. logarithmic convention with @dfielding14 in 2019, which led to me adding the paragraph you pasted to the Wiki.

[kurtsansom]

I need to go back over the derivation I got. I tried to write it out such that you don't need to store the locations in memory, but haven't been able to get that yet.