2019-RG-math-Vinogradov-translation

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Solve the problem of conjugating parts of the interval of integration in the following way. The vector P containing the physical parameters of the problem is formed using the

matrix M of coefficients and the required vector-function Y(x):

P MY (x)

where M is a square nondegenerate matrix.

Then at the conjugation point x = x* we can write expression

when passing through the conjugation point from the left to the right; index "-" means "to the left of the conjugation point", and the index "+" means "to the right of the conjugation point".

Suppose that the conjugation point does not coincide with the point of the orthogonal transformation. Then the expression for conjugation conditions of adjacent sections

will take the form

If now demand

then in the direct course of the sweep method, the integration can be continued from the left to the right in the following expressions:

Y матрица (x) M 1M Y- матрица (x) ,

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2.6. Properties of the transfer of boundary value conditions in S.K.Godunov’s sweep

method.

When solving a boundary-value problem for a system of "stiff" linear ordinary differential equations by S.K.Godunov’s method says that discrete orthogonalization is carried out by Gramm-Schmidt’s method with respect to vector-functions forming the variety of solutions of the given problem in order to overcome the tendency of degeneration of these vector-functions into linearly dependent ones.

At the same time, in the implementation of S.K.Godunov’s method, the boundary conditions from the initial boundary are also transferred to the other edge. Let us show the properties of this transfer.

Previously recorded

Then we can say that:

-the vector w*, which is unknown, is a vector of constants c,

-at the same time, the vector w* has the physical meaning of an external influence on the deformed system unknown at the edge x=0,

-the matrix W* is the matrix of boundary conditions unknown on the boundary x=0.

It follows from the formulated propositions that the transfer of boundary conditions in

S.K.Godunov’s method has the following meaning.

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means that the matrix of the boundary conditions W*, which are unknown at the edge x=0, is carried to the edge x=1.

Integration of differential equations is carried out with the goal of transferring the vector c to the edge x=1, and hence the vector w*, which expresses the conditions unknown at the edge

x=0.

The transfer of the matrix W* and the vector w* means that the matrix equation

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):

YL (x) Yматрица L (x)cL Y L (x) ,

YR (x) Yматрица R (x)cR Y R (x) .

At an arbitrary point we have

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Then we obtain

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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.

obtain:

That is, we obtain two matrix equations of boundary conditions transferred to the point x under consideration:

[UK(0 x)] Y (x) u - UY (0 x) ,

[VK(1 x)] Y (x) v VY (1 x) .

These equations are similarly combined into one system of linear algebraic equations with a square matrix of coefficients to find the solution Y (x) at any point x under

consideration:

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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:

physical parameters that do not enter into the parameters of the boundary conditions of the left edge U 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

The vector m 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 Y (0) is found from the expression:

MU Y (0) mu ,

that is, the vector Y (0) is found from the solution of a system of linear algebraic equations with a square non-degenerate coefficient matrix consisting of blocks U and M .

Similarly, we write on the right edge one more equation of the boundary conditions:

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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.

or

Thus, we have obtained an expression of the form:

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Hence we obtain that:

Thus, the required constants are found.