2.7. Modification of s.K.Godunov’s sweep method.
The solution in S.K.Godunov’s method is sought, as written above, in the form of the formula
.
We can write this formula in two versions - in one case the formula satisfies the boundary conditions of the left edge (index L), and in the other - the conditions on the right edge (index R):
,
.
At an arbitrary point we have
.
Then we obtain
,
,
.
That is, a system of linear algebraic equations of
the traditional kind with a square matrix of coefficients
for the computation of the vectors of
constants
is obtained.
Chapter 3. The method of "transferring of boundary value conditions" (the direct version of the method) for solving boundary value problems with non-stiff ordinary differential equations.
It is proposed to integrate by the formulas of the theory of matrices [Gantmakher] immediately from some inner point of the interval of integration to the edges:
,
.
We substitute the formula for
in the boundary conditions of the left edge and obtain:
,
,
.
Similarly, for the right boundary conditions, we obtain:
,
,
.
That is, we obtain two matrix equations of
boundary conditions transferred to the point
under consideration:
,
.
These equations are similarly combined into one
system of linear algebraic equations with a square matrix of
coefficients to find the solution
at any point
under consideration:
.
Chapter 4. The method of "additional boundary value conditions" for solving boundary value problems with non-stiff ordinary differential equations.
Let us write on the left edge one more equation of the boundary conditions:
.
As matrix
rows, we can take those boundary conditions, that is, expressions of
those physical parameters that do not enter into the parameters of
the boundary conditions of the left edge
or are linearly independent with
them. This is entirely possible, since for boundary value problems
there are as many independent physical parameters as the
dimensionality of the problem, and only half of the physical
parameters of the problem enter into the parameters of the boundary
conditions.
That is, for example, if the problem of the shell
of a rocket is considered, then on the left edge 4 movements can be
specified. Then for the matrix
we can take the parameters of forces
and moments, which are also 4, since the total dimension of such a
problem is 8.
The vector
of the right side is unknown and it
must be found, and then we can assume that the boundary value problem
is solved, that is, reduced to Cauchy’s problem, that is, the
vector
is found from the expression:
,
that
is, the vector
is found from the solution of a
system of linear algebraic equations with a square non-degenerate
coefficient matrix consisting of blocks
and
.
Similarly, we write on the right edge one more equation of the boundary conditions:
,
where
the matrix
is written from the same
considerations for additional linearly independent parameters on the
right edge, and the vector
is unknown.
For the right edge, too, the corresponding system of equations is valid:
.
We write
and substitute it into the last
system of linear algebraic equations:
,
,
,
.
We write the vector
through the inverse matrix:
and substitute it in the previous formula:
Thus, we have obtained a system of equations of the form:
,
where
the matrix
is known, the vectors
and
are known, and the vectors
and
are unknown.
We divide the matrix
into 4 natural blocks for our case and obtain:
,
from which we can write that
Consequently, the required vector
is calculated by the formula:
And the required vector
is calculated through the vector
:
,
.
Chapter 5. The method of "half of the constants" for solving boundary value problems with non-stiff ordinary differential equations.
In this method we use the idea proposed by S.K.Godunov to seek a solution in the form of only one-half of the possible unknown constants, but a formula for the possibility of starting such a calculation and further application of matrix exponents (Cauchy’s matrices) are proposed by A.Yu.Vinogradov.
The formula for starting calculations from the left edge with only one half of the possible constants:
,
.
Thus, a formula is written in the matrix form for the beginning of the calculation from the left edge, when the boundary conditions are satisfied on the left edge.
Then write
and
collectively:
,
and substitute in this formula the expression for Y(0):
or
.
Thus, we have obtained an expression of the form:
,
where
the matrix
has a dimension of 4x8 and can be
naturally represented in the form of two square blocks of 4x4
dimension:
.
Then we can write:
.
Hence we obtain that:
.
Thus, the required constants are found.
Chapter 6. The method of "transferring of boundary conditions" (step-by-step version of the method) for solving boundary value problems with stiff ordinary differential equations.