[PSyAd] [LFRic] Errors in use of LBOUND and looping in generated adjoint kernel

[tom-j-h]

Found when using PSyAD to generate adjoint kernel for science/gungho/source/kernel/solver/apply_mixed_operator_kernel_mod.F90 for LFRic Apps ticket 669.

The relevant bits of the required patch file for the generated adjoint kernel are:

-        u_e(idx_12 - iw3 + LBOUND(u_e(:,:), 1),df) = u_e(idx_12 - iw3 + LBOUND(u_e(:,:), 1),df) + q32(idx_12 + ij - iw3,1,df) * &
+        u_e(idx_12 - iw3 + LBOUND(u_e, dim=1),df) = u_e(idx_12 - iw3 + LBOUND(u_e, dim=1),df) + q32(idx_12 + ij - iw3,1,df) * &

-          u_e(idx_10 - iw2v + LBOUND(u_e(:,:), 1),df2) = u_e(idx_10 - iw2v + LBOUND(u_e(:,:), 1),df2) + mu_cd(idx_10 + ij - &
+          u_e(idx_10 - iw2v + LBOUND(u_e, dim=1),df2) = u_e(idx_10 - iw2v + LBOUND(u_e, dim=1),df2) + mu_cd(idx_10 + ij - &

-    do idx_7 = UBOUND(t_col, dim=1), LBOUND(t_col, dim=1), -1
-      t_col(:) = t_col(:) + mt_lumped_inv(idx_7 + iwt - LBOUND(t_col(idx_7), 1)) * t_col(idx_7)
-      t_col(idx_7) = 0.0
-    enddo
+    t_col(:) = t_col(:) * mt_lumped_inv(iwt:iwt+1+nm1)

-        u_e(idx_6 + LBOUND(u_e(:,:), 1) - 1,df) = u_e(idx_6 + LBOUND(u_e(:,:), 1) - 1,df) + (-pt2(idx_6 + ij - 1,2,df) * &
+        u_e(idx_6 + LBOUND(u_e, dim=1) - 1,df) = u_e(idx_6 + LBOUND(u_e, dim=1) - 1,df) + (-pt2(idx_6 + ij - 1,2,df) * &

-          u_e(idx_3 - iw2h + LBOUND(u_e(:,:), 1),df2) = u_e(idx_3 - iw2h + LBOUND(u_e(:,:), 1),df2) + mu_cd(idx_3 + ij - &
+          u_e(idx_3 - iw2h + LBOUND(u_e, dim=1),df2) = u_e(idx_3 - iw2h + LBOUND(u_e, dim=1),df2) + mu_cd(idx_3 + ij - &

There appears to be two problems here:

1. The lines starting with the variable u_e are calling LBOUND on an array section rather than an array, so this always returns 1 (https://gcc.gnu.org/onlinedocs/gfortran/LBOUND.html) and the adjoint produced was incorrect.
2. The idx_7 loop: firstly the LBOUND statement within the loop does not compile since it's being called on a scalar, but even so, the entire loop is unnecessary; the forward code contains the line:
   t_col(:) = t_col(:) * mt_lumped_inv(iwt:iwt+1+nm1)
   and since t_col is an active variable but mt_lumped_inv is a constant, the adjoint should contain the identical line (as seen in the patch).