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It should be noted that the differential equations sets mentioned above
and related to the time variable
t
, are the sets of conventional differential
equiations of the first order, which can be solved using a vector version of
the Runge-Kutta multistep method (in the present paper, a four-step version
of the method [3] is used, which has the fourth order of approximation
relative to time
t
)
.
We bring the vector version of the Euler’s equations set to a normal
form with a time derivative defined in the left part
∂ ~U
i
∂t
:
∂ ~U
i
∂t
=
L ~U
i
,
where
L
is the right part of the Euler’s equations set that does not contain
time derivatives. To a first approximation we use the solution obtained at
the previous time step. Then the four-step version of Runge-Kutta method
can be implemented in the form of the following sequence of steps:
~U
(1)
i
=
~U
(0)
i
+
Δ
t
4
L ~U
(0)
i
,
~U
(2)
i
=
~U
(0)
i
+
Δ
t
3
L ~U
(1)
i
,
~U
(3)
i
=
~U
(0)
i
+
Δ
t
2
L ~U
(2)
i
,
~U
(4)
i
=
h
~U
(0)
i
+ Δ
tL ~U
(3)
i
i
.
It is known that this way of searching for a solution
~U
i
relative to
t
,
solves one of the problems of the Euler’s equations numerical solution ,
i.e. the need to ensure the positivity of the required functions (if at the time
instant
t
n
the solution is positive, it remains positive at the time instant
t
n
+1
as well).
The rise of the approximation order relative to time variable
t
of the
Euler’s equations system numerical solution to the fourth order
O
(Δ
t
4
)
and higher is also possible if the sequence of meshes by time variable
t
and
extrapolation by limit offered by Richardson will be used. The Richardson
extrapolation has the following specific features:
•
the possibility to use the simpliest approximations of differential
problems;
•
the uniformity of algorithms implementation on the sequence of
meshes with different parameters of approximation;
•
the simplicity of the algorithm realization in general.
8 ISSN 0236-3941. HERALD of the BMSTU. Series “Mechanical Engineering”. 2014. No. 1