18.2.1 Solving the Linear System

The discretized scalar transport equation (Equation  18.2-2) contains the unknown scalar variable $\phi$ at the cell center as well as the unknown values in surrounding neighbor cells. This equation will, in general, be non-linear with respect to these variables. A linearized form of Equation  18.2-2 can be written as

where the subscript $nb$ refers to neighbor cells, and $a_P$ and $a_{\rm nb}$ are the linearized coefficients for $\phi$ and $\phi_{\rm nb}$.

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 which is described in Section  18.6.3.