The right hand side is constructed by adding, in succession, 1-D contributions coming from different sweeps:
It is called once for every direction at every integration stage. The single contribution, say for the
direction, is computed in a quasi-conservative way as
where the indices
and
stay fixed,
is the cell volume,
is the Riemann solver flux (not including pressure terms),
is the interface area. The source terms
are, respectively, the geometrical, gravitational and Powell's source terms (only for the MHD and RMHD modules).
The pressure term is separately discretized as a gradient operator and therefore it does not contain area factors nor it appears in the geometrical source terms. For instance, while incrementing the component of
corresponding to the normal momentum
we add
where
is a line element.
The computation of the right hand side closely reflects the nature of the divergence or gradient operators involved in the original equations. Indeed, for scalar quantities such as density or energy, one simply has
For vector quantities such as momentum and magnetic field, we exploit the symmetric or antisymmetric properties of the corresponding flux tensor.
(as in the momentum equation), we use the angular momentum-conserving form:
(as for the induction equation) the conservation law takes the form
.In spherical coordinates, for a symmetric tensor we use
(here
) whereas, if the tensor is antisymmetric,
, one has
The geometrical source terms (right hand sides in the previous equations) are evaluated by averaging interface values at the cell-centered.
1.8.2