Documentation/ProgrammerGuide/Centering

A final, important characteristic of the Mesh used in Athena is the centering of the conserved variables in Grid cells. The figure below shows this centering. The volume-averaged conserved variables $U$ = (density, momentum, total energy) are stored at cell-centers. The components of the area-averaged magnetic field are stored at the corresponding cell faces. Additional copies of the volume-averaged cell-centered magnetic field are stored in $U$, however these are not the fundamental description of the field (they are computed from the face-centered fields).

The cell-centered variables are stored in the Conserved Variable structure, which is a 3D array in the Grid structure. Components of this array are addressed using integer indices with the $i$ (x1) index incremented fastest: $U_{k,j,i}$.

The face centered magnetic fields are stored as 3D arrays directly in the Grid structure. The array element on the left-interface is given the index $i$, that is $B_{x,i-1/2,j,k} = B1i[k][j][i]$.