
Convergence and consistency analyses 185
As per Taylor series expansion
xt tt t
t
Ot
mn mttt
(, )( )
()
2
,
3
φ=φ−=φ−φ+φ−
−
(4.77)
xt tt t
t
Ot
mn nttt
(, )( )
()
2
,
3
φ=φ+=φ+φ+φ+
+
(4.78)
where
(,)
. We then substitute Equations (4.76) to (4.78) into
Equation (4.75) and simplify it to
xt
Ot
mn
t
()
,
2
=φ + (4.79)
Substituting Equation (4.74) and Equation (4.79) into Equation (4.67)
results in
PhtOta Oh
(,)(
)
,
2
,
2
φ=φ+ +φ+
(4.80)
Subtracting Equation (4.80) from Equation (4.66) gives
PPht aOta aOh
tx
(,
)
,,,
2
,
2
φ− φ=φ+φ−φ− −φ− (4.81)
Equation (4.81) tends to zero as h and t→ 0, which means that the pre-
sented discretisation scheme is consisten ...