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§2.3 spatial semidiscretization: a cell-node mfd method 21
• • •
(i, j + 1)
(i + 1, j + 1)
⊗
⊗
• •⊙
⊗
(i, j)
⊗
(i + 1, j)
• •
•
Figure 2.6: Stencil for the discrete operator A located at the internal cell center
(i + 1/2, j + 1/2). Symbols: • : Ψ h ·,·; ⊗ : (G X Ψ h )·,·, (G Y Ψ h )·,·, K XX
·,· , K XY Y Y
·,· , K·,· ;
•⊙ : (A Ψ h )i+1/2,j+1/2.
hand, the components of G Ψ h at the internal (i, j)-node reduce to:
(G XΨh )i,j = (Ψh i−1/2,j−1/2 + Ψh i−1/2,j+1/2 ) − (Ψh i+1/2,j−1/2 + Ψh i+1/2,j+1/2 )
2hX ,
2hY ,
(2.31)
for i = 2, 3, . . . , Nx − 1 and j = 2, 3, . . . , Ny − 1. The obtained formulae give an
approximation to the partial derivatives ∂xψ and ∂yψ, respectively. By contrast,
the expressions for the discrete gradient on the boundary nodes involve one-sided
differences. For instance, the x-component of G Ψh on the bottom boundary is
defined as:
(G Y Ψ h )i,j = (Ψh i−1/2,j−1/2 + Ψh i+1/2,j−1/2 ) − (Ψh i−1/2,j+1/2 + Ψh i+1/2,j+1/2 )
(G X Ψ h )i,1 = Ψh i−1/2,3/2 − Ψh i+1/2,3/2
hX ,
for i = 2, 3, . . . , Nx−1. Similar equations can be derived for the other boundaries.
Finally, at the left bottom corner, we get:
(G XΨh )1,1 = − Ψh 3/2,3/2
1 .
2
hX
Formulae for the remaining corner nodes are deduced analogously.
In order to obtain the formula for the discrete diffusion operator A = DK G ,
we shall apply the discrete divergence (2.30) to a vector W h whose components
at the (i, j)-node are given by:
W X i,j
W Y i,j
= KXX i,j (G XΨh )i,j + KXY i,j (G Y Ψh )i,j
= KXY i,j (G XΨh Y Y
)i,j + Ki,j (G Y Ψh )i,j,
•
(2.32)