114
5 2D Shallow-Water Modelling
where h is layer thickness. The discretised form of the diffusion term for u has the
components:
A h
h
∂
∂x
h
∂u
∂x
=
A h
h u Δx
⎡
⎣
h e
u
n
j,k+1 − u
n
j,k
Δx
−
h w
u
n
j,k − u
n
j,k−1
Δx
⎤
⎦ (5.34)
A h
h
∂
∂y
h
∂u
∂y
=
A h
h u Δy
⎡
⎣
h n
u
n
j+1,k − u
n
j,k
Δy
−
h s
u
n
j,k − u
n
j−1,k
Δy
⎤
⎦ (5.35)
where layer thicknesses of the faces of the control volume (see Fig. 5.16) are given
by:
h u = 0.5
h
n
j,k + h
n
j,k+1
h e = h
n
j,k+1
h w = h
n
j,k
h n = 0.25
h
n
j,k + h
n
j,k+1 + h
n
j+1,k + h
n
j+1,k+1
h s = 0.25
h
n
j,k + h
n
j,k+1 + h
n
j−1,k + h
n
j−1,k+1
The diffusion terms for v are constructed in a similar fashion.
Fig. 5.16 Illustration of locations of layer-thickness h and u grid points for the discretisation of
the horizontal diffusion terms for u