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25.2.1 Solving the Linear System

The discretized scalar transport equation (Equation  25.2-2) 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  25.2-2 can be written as

 a_P \, \phi = \sum_{\rm nb} \, a_{\rm nb} \phi_{\rm nb} + b (25.2-3)

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 grid 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 grid. This results in a set of algebraic equations with a sparse coefficient matrix. For scalar equations, 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  25.6.3.

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