contravariant velocities
Contravariant velocities are normally used in the pressure based finite volume methods (staggered and colocated grids). This looks more appropriate especially while applying boundary conditions like inlet, exit, wall, symmetry and periodic for irregular geometries.
Generally contravariant velocities are not (mentioned) used in density based finite volume methods. Why?. For irregular geometries, is it not more meaningful to use contravariant velocities while applying the boundary conditions. Or is it done but never mentioned. What is the physical meaning of contravariant velocity I will be extremely happy if these can be clarified Thank you John 
Re: contravariant velocities
>> Contravariant velocities are normally used in the pressure based finite volume methods (staggered and colocated grids).
Name a code which solves for contravariant velocity components? I suspect there are very, very few. Using contravariant velocity components makes little sense for colocated velocity components. The adoption of contravariant or covariant velocity components (usually scaled to be physical) is principally a means to use grid staggering to avoid pressure smoothing. There can be other advantages like the covariant form has no pressure gradient cross terms. What tends to kill the approach is the complexity associated with large number of terms and a lot of interpolation. >> Generally contravariant velocities are not (mentioned) used in density based finite volume methods. Why?. Since they almost always solve for Cartesian velocity components why should they mention them? >> For irregular geometries, is it not more meaningful to use contravariant velocities while applying the boundary conditions. Or is it done but never mentioned. Expressed in what coordinate system? >> What is the physical meaning of contravariant velocity Any reasonable maths book covering tensor calculus will explain the meaning of terms such as covariant and contravariant bases. (and you will find that, strictly, there is no such thing as a contravariant velocity) 
Re: contravariant velocities
Thank you Andy for you clarifications

Multigrid methods
Hi,
I was curious to know why Multi grid methods cannot be applied to hyperbolic equations and is applicable only for elliptic equations. Regards, Amith 
Re: Multigrid methods
Dear Mr Amith Multigrid methods are applied to both elliptic and hyperbolic equations. Please refer some of A.Jameson's papers on the multigrid methods applied to hyperbolic equations John

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