2.5. Conjugation of parts of the integration interval for s.K.Godunov’s sweep method.

To automate the computational process on the entire integration interval, which is composed for conjugate shells with different physical and geometric parameters, the deformation of which is described by different functions, it is necessary to have the procedures of conjugation of the corresponding functions.

In the general case, the resolving functions of various parts of the integration interval of the problem have no physical meaning, and the physical parameters of the problem are expressed in various ways through these functions and their derivatives. At the same time, conjugation of adjacent sections must satisfy kinematic and force conditions at the point of conjugation.

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):

where M is a square nondegenerate matrix.

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

_,

where is the vector corresponding to a discrete change in the physical parameters 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".

In S.K.Godunov’s method, the vector-function of the problem on each section is sought in the form

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:

_,

_.

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

и .

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.

The continuation of integration, beginning with the vector , means the transfer of the "convolution" of the matrix equation of the boundary conditions at x=0 to the right edge x=1.

The continuation of the integration, beginning with the vectors in the matrix , 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 of the boundary conditions, which are unknown at the edge x=0, is carried to the edge x=1.