[Mitgcm-support] Re: upwind

[mitgcm-support at dev.mitgcm.org]

Arne Biastoch wrote:
> (i) how do you get the numbers -0.125 and 0.5? When I look at the 3.
> order
> upwind discretization (e.g. 2.28 in Haidvogel and Beckmann), there are
> factors of -1/3 and 4/3. Even if you consider the divisions by 2dx and
> 4dx
> the remaining factors are -1/12 and 2/3.

Assuming U>0 then the stencil for the right hand flux is
composed of centered average (4/8=1/2=0.5)
   [  0    0   4/8  4/8 ]
and a side ways diffusion (1/8=0.125)
 + [  0  -1/8  2/8 -1/8 ]
which gives a total stencil of
 = [ 0   -1/8  6/8  3/8 ]  = R
Similarly, the left hand flux has a stencil of
   [  0   4/8  4/8   0  ]
 + [-1/8  2/8 -1/8   0  ]
 = [-1/8  6/8  3/8   0  ]  = L
If we now take the difference between the fluxes (div.F)
and add up the stencils we get
R-L = 
   [1/8  -7/8  3/8  3/8 ]
which is what is written out in 2.27 in Haidvogel and Beckmann.

Eq. 2.28 is fourth order which can be written as a
centered flux - a centered evaluation of d_xx times 1/6.
  R = [  0     0   6/12  6/12 0 ] + [   0   -1/12 1/12  1/12 -1/12 ]
  L = [  0    6/12 6/12   0   0 ] + [ -1/12  1/12 1/12 -1/12   0   ]
R-L = [ 1/12 -8/12  0    8/12 -1/12 ]
We write things like this so we can evaluate boundary conditions
more naturally.

> (ii) for the case uTrans < 0 TxM is always zero.
>            TxM=(theta(i-1,j,k,bi,bj)-theta(i-1,j,k,bi,bj))*
>      &          _maskW(i,j,k,bi,bj)

bug fix:
             TxM=(theta(i,j,k,bi,bj)-theta(i-1,j,k,bi,bj))*
       &          _maskW(i,j,k,bi,bj)

Alistair.
cc support