FABM advection with semi-implicit and TVD

[riquitorres]

Submitted by Karsten Lettmann

1. We had some problems with advection of FABM tracers when activating
   with semi-implicit and TVD.
   The tracers were not advected in this case.

For normal explicit runs using TVD everything was fine.

I compared with the advection of salt and there might be some
inconsistency in the code.
I checked your FABM 4.3 and FABM 4.4.1 code I downloaded a few months,
ago. Both had the same issue.

I believe, the problem is in mod_fabm_3D.F in SUBROUTINE FABM_ADV in the
lines (1675 in commit ):

#       if !defined (SEMI_IMPLICIT)
#       if !defined (TVD)
         EXFLUX=-UN*DTIJ(I,K)* &
((1.0_SP+SIGN(1.0_SP,UN))*FIJ2+(1.0_SP-SIGN(1.0_SP,UN))*FIJ1)*0.5_SP+FXX+FYY
#       else
         EXFLUX=-UN*DTIJ(I,K)* &
((1.0_SP+SIGN(1.0_SP,UN))*FAB12_B+(1.0_SP-SIGN(1.0_SP,UN))*FAB12_A)*0.5_SP+FXX+FYY
#       endif
#       else
         EXFLUX=-UN*DTIJ(I,K)* &
((1.0_SP+SIGN(1.0_SP,UN))*FIJ2+(1.0_SP-SIGN(1.0_SP,UN))*FIJ1)*0.5_SP
EXFLUX=(1.0_SP-IFCETA)*EXFLUX+IFCETA*(-UN1*DTIJ1(I,K)*((1.0_SP+SIGN(1.0_SP,UN1))*FIJ2+(1.0_SP-SIGN(1.0_SP,UN1))*FIJ1)*0.5_SP)+FXX+FYY
#       endif

For a semi-implicit run the following lines might be used even for TVD:

         EXFLUX=-UN*DTIJ(I,K)* &
((1.0_SP+SIGN(1.0_SP,UN))*FIJ2+(1.0_SP-SIGN(1.0_SP,UN))*FIJ1)*0.5_SP
EXFLUX=(1.0_SP-IFCETA)*EXFLUX+IFCETA*(-UN1*DTIJ1(I,K)*((1.0_SP+SIGN(1.0_SP,UN1))*FIJ2+(1.0_SP-SIGN(1.0_SP,UN1))*FIJ1)*0.5_SP)+FXX+FYY

Here, the variables FIJ1 and FIJ2 are used, which are not defined in
TVD, where you only specify:

        FAB12_A     =
FABM_ALL(IA,K,IFABM)+0.5_SP*A_LIMITER*(FABM_ALL(IB,K,IFABM)-FABM_ALL(IA,K,IFABM))
        FAB12_B     =
FABM_ALL(IB,K,IFABM)+0.5_SP*B_LIMITER*(FABM_ALL(IA,K,IFABM)-FABM_ALL(IB,K,IFABM))

When comparing it with the adv_s, an easy and compact solution might be:

        FIJ1     =
FABM_ALL(IA,K,IFABM)+0.5_SP*A_LIMITER*(FABM_ALL(IB,K,IFABM)-FABM_ALL(IA,K,IFABM))
! FAB12_A
        FIJ2     =
FABM_ALL(IB,K,IFABM)+0.5_SP*B_LIMITER*(FABM_ALL(IA,K,IFABM)-FABM_ALL(IB,K,IFABM))
! FAB12_B

#       if !defined (SEMI_IMPLICIT)
         EXFLUX=-UN*DTIJ(I,K)* &
((1.0_SP+SIGN(1.0_SP,UN))*FIJ2+(1.0_SP-SIGN(1.0_SP,UN))*FIJ1)*0.5_SP+FXX+FYY
#       else
         EXFLUX=-UN*DTIJ(I,K)* &
((1.0_SP+SIGN(1.0_SP,UN))*FIJ2+(1.0_SP-SIGN(1.0_SP,UN))*FIJ1)*0.5_SP
EXFLUX=(1.0_SP-IFCETA)*EXFLUX+IFCETA*(-UN1*DTIJ1(I,K)*((1.0_SP+SIGN(1.0_SP,UN1))*FIJ2+(1.0_SP-SIGN(1.0_SP,UN1))*FIJ1)*0.5_SP)+FXX+FYY
#       endif