where is a unit vector that is normal to the face at each point on the face.
where || denotes the magnitude (standard Euclidian norm) of .
where a superscript ``t'' denotes ``transpose.'' The three face-area components associated with the Right-Top-Up vertex are illustrated in Fig.4. The algebraic flux vectors for the other vertices are defined analogously.
where f is the face index and the sum is taken over all faces.
where v is the vertex index, and Vv denotes the volumetric weight associated with vertex v .
where i, j denotes the vector from vertex i to vertex j .
where denotes a Left-Bottom-Down Cartesian flux vector,
denotes a Left-Bottom-Down face-area flux vector,
where aLx denotes the x-component of the area vector associated with the left face. The remaining components of the matrix are defined analogously.
and the dot product is taken in the usual way.
where the sum is taken over all faces.
while requiring continuity of the flow yields:
where V denotes the total cell volume, QC denotes the cell-center inhomogeneous source, and the cell index (i, j, k) has been supressed for simplicity.
where L1, j, k is the Left area vector associated with cell (1, j, k) , and is the extrapolated intensity associated with the boundary face.