[Techies]

Hi all,
I read the document about the cfd, they said that the uniform spacing mesh has rectangular mesh with second order accuracy and the nonuniform mesh (the changes of mesh take place within domain)is only of first order accuracy
So the uniform mesh is better than non uniform mesh ? Is this true ?

[PenPencil]

No. This is a much more complex matter. Second-order accuracy can be obtained with nonuniform meshes, thank goodness. Otherwise we would have a very big problem to refine any domain from the wall outwards.

[FMDenaro]

The type of computational mesh has nothing to do with the order of accuracy. You have to check the local truncation errro of the discretization to talk about accuracy.
On non-uniform grid you can get any accuracy order, provided that the scheme is suitably built.

[naffrancois]

As suggested by FMDenaro this topic is strongly related to the truncation error of the spatial scheme.

For a given grid nodes number and a consistent scheme (at least 1st order), uniform spacing will not give you the lowest global error.

The reason is that minimum overall error is reached when truncation error is minimized. Truncation error is however never constant over the domain as it depends also on higher derivatives of the solution. In an attempt to equi-distribute/minimize truncation error grid points have to be moved.

[andy_]

Originally Posted by Techies

So the uniform mesh is better than non uniform mesh ? Is this true ?

Quite often yes but as a general blanket statement no.

If what you are solving pushes the approach towards being close to uniform everywhere then a uniform grid with a high order numerical scheme is likely to be the most efficient approach. Acoustics and DNS/LES turbulence can be examples of this.

If what you are solving has a few strong local gradients with very much smaller ones throughout the rest of the flow then this pushes the approach towards a relatively low order scheme with strong variations in grid refinement. The low order scheme follows from high order schemes generally misbehaving badly when resolution is insufficient to resolve the gradients in the solution. Shock waves and free/boundary layers using RANS turbulence models can be examples of this.

As a final point. The order of the numerical scheme matters most when trying to drive the truncation error to negligible levels rather than living with a significant level and seeking to make the form more interpretable and/or better behaved physically (e.g. numerical diffusion). Consider a grid refinement study (e.g. sequence of grids halving the grid spacing everywhere) for a numerical scheme that is formally second-order accurate on a uniform grid and formally first-order accurate on a non-uniform grid. If the non-uniform grid is varied smoothly and well matched to the unresolved gradient (e.g. boundary layer) then the observed convergence rate over the range down to practical grid independence may be more like 1.9 rather than 1.0 with the uniform grid being 2. With the global error for the non-uniform grid on the coarsest grid level starting well below that of the uniform grid due to it's smaller grid spacing next to the wall the non-uniform grid may well reach practical grid independence first (e.g. less than 1% error in relevant physical quantity). Of course if grid refinement is continued down towards round-off at some point the uniform grid will begin to have the lower global error but this is often largely irrelevant in real world simulations.