7.2 Choice of Variable Arrangement on the Grid
165
of numerical grids were outlined in Chap. 2. There are, however, many variants of the distribution of computational points within the solution domain.
The basic arrangements associated with the FD and FV discretization methods were shown in Figs. 3.1 and 4.1. These arrangements may become more
complicated when coupled equations for vector fields (like the Navier-Stokes
equations) are being solved. These issues are discussed below.
7.2.1 Colocated Arrangement
The obvious choice is to store all the variables a t the same set of grid points
and to use the same control volumes for all variables; such a grid is called
colocated, see Fig. 7.1. Since many of the terms in each of the equations
are essentially identical, the number of coefficients that must be computed
and stored is minimized and the programming is simplified by this choice.
Furthermore, when multigrid procedures are used, the same restriction and
prolongation operators for transfer of information between the various grids
can be used for all variables.
The colocated arrangement also has significant advantages in complicated
solution domains, especially when the boundaries have slope discontinuities
or the boundary conditions are discontinuous. A set of control volumes can be
designed to fit the boundary including the discontinuity. Other arrangements
of the variables lead to some of the variables being located at singularities of
the grid, which may lead to singularities in the discretized equations.
Fig. 7.1. Colocated arrangement of velocity components and pressure on a FD
(left) and FV (right) grid
The colocated arrangement was out of favor for a long time for incompressible flow computation due to the difficulties with pressure-velocity coupling and the occurrence of oscillations in the pressure. From the time the
staggered grid was introduced in the mid-1960s until the early 1980s, the
colocated arrangement was hardly used. Then, use of non-orthogonal grids
became more commonplace as problems in complex geometries began to be
tackled. The staggered approach can be used in generalized coordinates only
165
of numerical grids were outlined in Chap. 2. There are, however, many variants of the distribution of computational points within the solution domain.
The basic arrangements associated with the FD and FV discretization methods were shown in Figs. 3.1 and 4.1. These arrangements may become more
complicated when coupled equations for vector fields (like the Navier-Stokes
equations) are being solved. These issues are discussed below.
7.2.1 Colocated Arrangement
The obvious choice is to store all the variables a t the same set of grid points
and to use the same control volumes for all variables; such a grid is called
colocated, see Fig. 7.1. Since many of the terms in each of the equations
are essentially identical, the number of coefficients that must be computed
and stored is minimized and the programming is simplified by this choice.
Furthermore, when multigrid procedures are used, the same restriction and
prolongation operators for transfer of information between the various grids
can be used for all variables.
The colocated arrangement also has significant advantages in complicated
solution domains, especially when the boundaries have slope discontinuities
or the boundary conditions are discontinuous. A set of control volumes can be
designed to fit the boundary including the discontinuity. Other arrangements
of the variables lead to some of the variables being located at singularities of
the grid, which may lead to singularities in the discretized equations.
Fig. 7.1. Colocated arrangement of velocity components and pressure on a FD
(left) and FV (right) grid
The colocated arrangement was out of favor for a long time for incompressible flow computation due to the difficulties with pressure-velocity coupling and the occurrence of oscillations in the pressure. From the time the
staggered grid was introduced in the mid-1960s until the early 1980s, the
colocated arrangement was hardly used. Then, use of non-orthogonal grids
became more commonplace as problems in complex geometries began to be
tackled. The staggered approach can be used in generalized coordinates only