
Stability analyses 233
the constants in the above equation given as A = 0.5 + (0.5 a t a
i,i
) + (a t
a
i,i–1
) and B = 0.5 + (0.5 a t a
i,i
) + (a t a
i,i+1
), respectively, where a
i,i
, a
i,i–1
,
and a
i,i+1
are the DQ weighting coefcients. Following conditions are thus
derived for Equation (5.46) to be consistent, namely for P u – P(h
x
, h
y
) u →
0 to be true.
−
=
−−
=
+
+
≥
≥+
Bh Ah
at
AB
t
Ah Bh
t
at
1,
1
0, and
0
0for ve or zerodissipation
21 1
2
2
22
(5.47)
for +ve or zero dissipation must be satised. Equation (5.47) provides the
constraints that, when satised, result in the consistent discretisation of the
wave equation by the uniform and cosine distributions ...