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86 Numerical solutions of the equations
Fully staggered 2D grid
V
x
V
y
P
,
y
rh
Fig. 7.7 Example of a fully staggered 2D numerical grid.
in case of constant viscosity when unknown parameters are components of velocity
(v
x
, v
y
) defined at the basic nodes, and pressure (P) defined at the additional nodes
(Fig. 7.6). Accordingly, x-andy-Stokes equations (Eq. 5.22) are formulated at
basic nodes, whilst the continuity equation is formulated at the additional nodes.
Yet, half-staggered grids are less natural for mechanical and thermomechanical
problems with variable viscosity.
Fully staggered grids are applied in two and three dimensions and consist of a
combination of several types of nodal points having different geometrical positions
(Fig. 7.7). Different variables are then defined at different nodal points. Differ-
ent equations are also formulated at different nodal points. Despite the apparent
geometrical complexity, fully staggered grids are the most convenient choice for
thermomechanical numerical problems with variable viscosity when finite dif-
ferences are used for solving the continuity, Stokes and temperature equations.
Discretisation of thermomechanical equations on the fully staggered grid is very
natural and gives simple FD formulas. Also, the accuracy of a numerical solution
obtained on a fully staggered grid is notably (up to four times, Fornberg, 1995)
higher then that on a non-staggered grid. Therefore: ‘think in a fully staggered
way!’ (I’ve done this since 2002).
7.2 Discretisation of the equations
The discretisation of the equations depends on the type of numerical grid that
is employed. The following discretisation schemes can for example be used to
describe the 2D incompressible continuity equation
∂v
x
∂x
+
∂v
y
∂y
= 0 in the case of
non-staggered (half-staggered) and a fully staggered 2D grid.
Non-staggered (half-staggered) grid (Fig. 7.8):
v
x3
− v
x1
+ v
x4
− v
x2
2x
+
v
y2
− v
y1
+ v
y4
− v
y3
2y
= 0. (7.1)