The inviscid fluxes appearing in Eqn. 45 are evaluated by an upwind, flux-difference splitting scheme as described in [4] and [3]. This scheme acknowledges that 
contains characteristic information propagating through the domain with speed and direction according to the eigenvalues of the system. By splitting 
into parts, where each part contains information travelling in a particular direction (that is, characteristic information), and upwind differencing the split fluxes in a manner consistent with their corresponding eigenvalues, the following expression for the value of the flux at each face is obtained:
| (46) |
where the subscripts "0" and "1" refer to the cells on either side of face-f, and 
is given by:
| (47) |
where 
and 
are the solution vectors from cell-0 and cell-1 interpolated to the face using reconstruction gradients.
The matrix 
is defined by:
| (48) |
where 
is the diagonal matrix of eigenvalues and 
is the modal matrix that diagonalizes 
.
By using these reconstructed solution vectors, the discretization scheme becomes formally second-order accurate. In this form, Eqn. 46 can be viewed as a second-order central-difference plus an added matrix dissipation. The added matrix dissipation term is not only responsible for producing an upwinding of the convected variables, and of pressure and flux velocity in supersonic flow, but it also provides the pressure-velocity coupling required for stability and efficient convergence of low-speed and incompressible flows.