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Let us take the solution to the problem in question obtained by

intergating (with the second order of approximation

O

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

t

, and also the solution (with the second order of

approximation

O

t

2

))

, denoted as

(

ρ, u, v, P

)

|

t

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

t

=

4

3

(

ρ, u, v, P

)

|

b

t

t

t/

2

1

3

(

ρ, u, v, P

)

|

b

t

t

t

brings the exact solution nearer to the approximation of the fourth order

relative to time variable

O

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

t

=

32

21

(

ρ, u, v, P

)

|

b

t

t

t/

4

4

7

(

ρ, u, v, P

)

|

b

t

t

t/

2

+

1

21

(

ρ, u, v, P

)

|

b

t

t

t

;

(

ρ, u, v, P

)

|

b

t

t

t

=

512

315

(

ρ, u, v, P

)

|

b

t

t

t/

8

32

45

(

ρ, u, v, P

)

|

b

t

t

t/

4

+

4

45

(

ρ, u, v, P

)

|

b

t

t

t/

2

1

315

(

ρ, u, v, P

)

|

b

t

t

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