7 / 21
Let us take the solution to the problem in question obtained by
intergating (with the second order of approximation
O
(Δ
t
2
))
relative
to time
t
with increment
Δ
t
at the time instant
b
t
=
t
+ Δ
t
, which we
denote as
(
ρ, u, v, P
)
|
t
+Δ
t
, and also the solution (with the second order of
approximation
O
(Δ
t
2
))
, denoted as
(
ρ, u, v, P
)
|
t
+Δ
t/
2
, obtained by using
two time increments (each increment is equal to
Δ
t/
2)
before the time
instant
b
t
=
t
+ Δ
t
.
Then the linear combination:
(
ρ, u, v, P
)
|
b
t
t
+Δ
t
=
4
3
(
ρ, u, v, P
)
|
b
t
t
+Δ
t/
2
−
1
3
(
ρ, u, v, P
)
|
b
t
t
+Δ
t
brings the exact solution nearer to the approximation of the fourth order
relative to time variable
O
(Δ
t
4
)
[4].
To bring the exact solution nearer to the sixth or eighth order of
approximation relative to time variable, the formulas [5] should be used:
(
ρ, u, v, P
)
|
b
t
t
+Δ
t
=
32
21
(
ρ, u, v, P
)
|
b
t
t
+Δ
t/
4
−
−
4
7
(
ρ, u, v, P
)
|
b
t
t
+Δ
t/
2
+
1
21
(
ρ, u, v, P
)
|
b
t
t
+Δ
t
;
(
ρ, u, v, P
)
|
b
t
t
+Δ
t
=
512
315
(
ρ, u, v, P
)
|
b
t
t
+Δ
t/
8
−
−
32
45
(
ρ, u, v, P
)
|
b
t
t
+Δ
t/
4
+
4
45
(
ρ, u, v, P
)
|
b
t
t
+Δ
t/
2
−
1
315
(
ρ, u, v, P
)
|
b
t
t
+Δ
t
.
At the first fractional step, the following divergent form of Euler’s
equiations is used:
∂ρ
∂t
+
∂ρu
ξ
∂ξ
=
F
ρ
,
∂
(
ρu
ξ
)
∂t
+
∂ ρu
2
ξ
+
P
∂ξ
=
F
ρu
,
∂
(
ρE
)
∂t
+
∂
(
ρEu
ξ
+
Pu
ξ
)
∂ξ
=
F
E
,
∂ ~U
∂t
+
∂F ~U
∂ξ
=
~F
2
,
where
u
ξ
= (
u, v
)
, parameter
ξ
can take one of the set of values
(
r, z
)
, the
solution vector is
~U
= (
ρ, ρu
ξ
, ρE
)
T
, the vector of the flow variable will
be written as
F ~U
=
ρu
ξ
, ρu
2
ξ
+
P, ρEu
ξ
+
Pu
ξ
T
, and the right part
vector will be given as
~F
2
= (
F
ρ
, F
ρu
, F
E
)
T
. Here (for the time fractional
step
t
∈
[
t, t
+ Δ
t/
2])
, the non-linear quasi-monotonous conpact difference
scheme of the higher order of accuracy is used, which in the space-smooth
ISSN 0236-3941. HERALD of the BMSTU. Series “Mechanical Engineering”. 2014. No. 1 9