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I'm trying to understand when and why one would use a staggered vs. a colocated grid in problems that have velocities and scalars that they transport (e.e. density).

If scalars are defined cell-centered, then in the physics world you often hear that velocities should be defined between cells, because apart from being intuitive, the resulting numerical schemes are stable.
I'm certain to having had heard this statement in the context of finite difference codes.

But now I'm learning finite volume methods with the Riemann Solvers by E. Toro. There, in the math world, vectors like velocities are just 3-scalars in equation systems that are all solved on one and the same colocated grid. There is never any mention of staggering grids at all (at least not as far as I got in the book, around Chapter 16, 3rd Ed.).

So is the choice of defining velocities on staggered vs. colocated grids related to the numerical methods one is using, or rather if one is solving ODEs vs. PDEs, or is the answer something completely different?

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    $\begingroup$ It's about the choice of numerical method: some methods are based on staggered grids, others on co-located meshes. I don't think that one is fundamentally better than the others, though it's generally the case that it's easier to derive higher order schemes with co-located meshes. $\endgroup$ – Wolfgang Bangerth Nov 10 '17 at 22:59
  • $\begingroup$ @WolfgangBangerth: Thank you for the comment. If you could provide a source maybe, then we could convert that into an answer and accept it! $\endgroup$ – AtmosphericPrisonEscape Nov 11 '17 at 13:07
  • $\begingroup$ It's such a general statement that I don't think there is a source. You'll find whole books about cell-/vertex-centered FD methods, and whole books about cell-based finite element methods. They're just different approaches to the same thing. $\endgroup$ – Wolfgang Bangerth Nov 12 '17 at 16:21
  • $\begingroup$ @WolfgangBangerth: I see. Then I may rephrase my question: Given a simple problem, let's say the 1D nonlinear Burgers equation. And I find both implementations for the velocity grid in two different codes. Can I say something meaningful about what the authors of those codes wanted to achieve? $\endgroup$ – AtmosphericPrisonEscape Nov 12 '17 at 16:45
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    $\begingroup$ Not really. I would say it probably says more about the authors academic background and education than about a conscious choice. If you were educated by finite difference people, you will choose a staggered mesh, whereas if you have a finite element background, you'll use a single mesh. $\endgroup$ – Wolfgang Bangerth Nov 12 '17 at 21:29

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