The discretized scalar transport equation (Equation 23–2) contains the unknown scalar variable at the cell center as well as the unknown values in surrounding neighbor cells. This equation will, in general, be nonlinear with respect to these variables. A linearized form of Equation 23–2 can be written as

(23–3)

where the subscript refers to neighbor cells, and and are the linearized coefficients for and .

The number of neighbors for each cell depends on the mesh topology, but will typically equal the number of faces enclosing the cell (boundary cells being the exception).

Similar equations can be written for each cell in the mesh. This results in a set of algebraic equations with a sparse coefficient matrix. For scalar equations, Ansys Fluent solves this linear system using a point implicit (Gauss-Seidel) linear equation solver in conjunction with an algebraic multigrid (AMG) method that is described in Algebraic Multigrid (AMG).