The namespaces fvm:: and fvc:: allow for the evaluation of a variety of operators implicitly and explicitly, respectively.

The explicit operator fvc, named “finite volume calculus,” returns an equivalent field based on the actual field values. For example, the operator fvc::div(φ) returns an equivalent geometric field in which each cell contains the value of the divergence of the variable φ.

In contrast, the fvm implicit operator defines the implicit finite volume discretization in terms of matrices of coefficients. For example, fvm::laplacian(φ) returns an fvMatrix in which all the coefficients are based on the finite volume discretization of the Laplacian.

The role of the fvm:: and fvc:: operators is to construct the LHS and RHS of a system of equations representing the discretized form of Eq. (5.43) over each element in the mesh. The discretization process yields a system of equations that can be represented in the matrix form shown in Fig. 5.19. Each row of the system represents the discretized equation in one element, with the off-diagonal coefficients representing the mutual influence of the neighboring elements.

In OpenFOAM®, the matrix of coefficients is stored in a class named lduMatrix and the specialized fvMatrix. The chosen storage format of the coefficients is based on the ldu sparse format, in which all coefficients are stored in three main arrays: diagonal, upper, and lower. The diagonal coefficients are thus stored in one array accessed by the diag() function with its dimension being equal to the number of elements in the computational domain. The upper and lower coefficients are stored each in an array as shown in Fig. 5.20. The source or the right-hand side is stored under a specific vector called source with its dimension being also equal to the number of cells over the computational domain.