Discrete Ordinates: follow the light along a few chosen directions

How the Discrete Ordinates radiation model works Four panels. One: the selected light pattern, how the hot cell emits. Two: the fixed set of directions the model solves along, drawn as equal blue arrows because they do not depend on the light, with an amber line along each showing the brightness read off the pattern. Three: every cell of the mesh carries the same equal blue direction arrows, and an amber line along each shows the brightness the solver finds there; cells that no beam reaches have no amber. Four: the radiation field the model predicts around a hot spot: with few directions it is a star of streaks, the ray effect; with many directions the streaks merge into a smooth glow. A mesh selector shows that a coarse mesh smears the field sideways (false scattering) while a fine one keeps it sharp. The true field is a dashed outline. 1. Selected light pattern how the hot cell emits brightness by direction dashed = reality, as in (4) 2. Model directions a fixed set, whatever the light 8 directions, all equal amber = brightness along each 3. Solving in every cell brightness found along each blue: directions · amber: result 8 transport solves 4. The result the radiation field dashed = reality · 32×32 cells
Try a different light pattern
How fine a mesh?
8
Fig. 4 — Discrete ordinates in one picture. Instead of following light in every direction, the method picks a handful of directions (2) and solves how light travels through the mesh along each one (3). The result (4) is the field it predicts around a hot spot, with reality as a dashed outline. With few directions the light can only travel along those few lines, so an even glow comes out as a star of streaks — the ray effect — and a beam is bent onto the nearest allowed direction. More directions smooth this out, at the price of one more transport solve per direction. The mesh matters too: on a coarse mesh light leaks sideways from cell to cell as it travels — false scattering — which blurs the streaks but also smears a beam; a fine mesh keeps things sharp and costs many more cells.
Why can 64 × 64 look worse than 32 × 32?

Two errors are at play. The ray effect comes from too few directions and does not care about the mesh. False scattering comes from the mesh: on coarse cells it smears the streaks into each other — which, for a diffuse glow, happens to push the picture toward the smooth truth. A fine mesh removes that blur and exposes the angular error underneath. It does not add error; it stops hiding it.

Refining the mesh alone never converges a DOM solution: you have to add directions and cells together. Try it — at 64 × 64, going from 8 to 32 directions does far more than going from 32 × 32 to 64 × 64. For a beam the trade-off runs the other way, since smearing is always harmful there.