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The space variables
∂Y
∂ξ
i,j
included in piecewise-polynomial distribu-
tions
Y
(
ξ
)
are computed as follows.
Firstly, for the discrete function
Y
i
we determine the approximate value
F
i
of the first partial derivative related to the space variable
ξ
with the
eighth order of approximation.
There to, in each mesh with number
i
for each value to be retrieved
Y
i,j
the index of non-monotony
Ind
(
Y
)
is to be computed:
Ind
(
Y
)
i
=
1
12
|−
Y
i
+2
,j
+ 16
Y
i
+1
,j
−
30
Y
i,j
+ 16
Y
i
−
1
,j
−
Y
i
−
2
,j
|
1
2
−
Y
i
+2
,j
+4
Y
i
+1
,j
−
3
Y
i,j
+
1
2
3
Y
i,j
−
4
Y
i
−
1
,j
+
Y
i
−
2
,j
+
θ
,
where
θ
is a small parameter.
Then we find the first derivative
f
by variable
ξ
according to the usual
approximation formula of the second order of approximation and fulfil its
“monotonous limitation” on the mesh:
Ind
(
Y
)
i
= 1
∙
Ind
(
Y
)
i
+ 2
∙
[1
−
Ind
(
Y
)
i
] ;
f
i
=
Y
i
+1
−
Y
i
−
1
2Δ
,
˜
f
i
=
sign
(
Y
i
+1
−
Y
i
−
1
) min
Ind
(
Y
)
i
+1
|
f
i
+1
|
,
|
f
i
|
, Ind
(
Y
)
i
−
1
|
f
i
−
1
|
,
where
Δ
is a step of the space mesh in direction
ξ
. Then the approximate
“monotonized” value of
˜
F
i
of the first partial derivative by the space
variables
ξ
with the approximation error on the level of
F
i
=
∂
∂ξ
+
Δ
6
2100
+
+
O
(Δ
8
)
can be obtained by the formula (i.e., by the solution of the system
of equations with the tridiagonal marix):
Q
i
=
E
+
Δ
2
30
˜
f
i
,
e
F
i
=
(
E
+
Δ
2
6
−
1
Q
i
)
i
;
F
i
=
sign
(
Y
i
+1
−
Y
i
−
1
) min ˜
F
i
+1
,
˜
F
i
,
˜
F
i
−
1
;
F
i
=
F
i
+
sign
(
Y
i
+1
−
Y
i
−
1
)
∙
[1
−
Ind
(
Y
)
i
]
∙
e
F
i
−
F
i
,
where
Δ
0
f
i
=
f
i
+1
−
f
i
−
1
,
Δ
2
f
i
=
f
i
+1
−
2
f
i
+
f
i
−
1
,
E
is a unit operator.
Note, that the given formula is the symmetrical finite difference of the sixth
order of approximation [7]. This form of computing of the first derivative
F
i
is used to form the edge conditions while obtaining the approximate
“monotonized” value
˜
F
i
of the first partial derivative by the space variables
ISSN 0236-3941. HERALD of the BMSTU. Series “Mechanical Engineering”. 2014. No. 1 11