2.2. The second algorithm for the beginning of the calculation by s.K.Godunov’s sweep method.

This algorithm requires the addition of a matrix of boundary conditions to a square non-degenerate one:

.

The initial values are found from the solution of the following systems of linear algebraic equations:

,

where is a vector of zeros of dimension 4х1.

The column vectors and the column vector are linearly independent and, taking part in the formation of the vector , satisfy the boundary condition .

2.3. The replacement of the Runge-Kutta’s numerical integration method in s.K.Godunov’s sweep method.

In S.K.Godunov's method, as shown above, the solution is sought in the form:

.

At each specific section of S.K.Godunov's method of sweeping between points of orthogonalization one can use the theory of matrices instead of Runge-Kutta’s method and perform the calculation through Cauchy’s matrix:

.

So perform calculations faster, especially for differential equations with constant coefficients, since in the case of constant coefficients it is sufficient to calculate once Cauchy’s matrix in a small section and then only multiply by this once computed Cauchy’s matrix.

Similarly, through the theory of matrices, we can also calculate the vector of a particular solution of an inhomogeneous system of differential equations. Or, for this vector, Runge-Kutta’s method can be used separately, that is, one can combine the theory of matrices and Runge-Kutta’s method.

2.4 Matrix-block realizations of algorithms for starting calculation by s.K.Godunov’s sweep method.

We consider a system of linear algebraic equations expressing the boundary conditions for x=0

Suppose that there is a constructed quadratic nondegenerate matrix .

Similarly, we write a vector where the introduced vector is unknown.

We write the system of linear algebraic equations

_{or in block form}

_.

It follows that

_.

Imagine .

Then

_.

At the same time, we remember that the solution of the boundary value problem is sought in the form

_.

Comparing

and

given that here the vector of unknown constants is , we obtain the initial values of the vectors for the beginning of integration in S.K.Godunov’s method.:

и .

Another matrix derivation can be stated in the following form.

We transform the system

_{by line orthonormalization to an equivalent system with orthonormal rows}

_.

Then we can write

_.

Making a comparison of two expressions:

and from what is a vector of unknown constants, we get:

и .

Note that another matrix-block derivation of the formulas is possible.

Transition from the system

to the system

can be realized by another method, replacing the line orthonormation of by the following orthonormal decomposition of the matrix G

where the matrix J has orthonormal columns, and the matrix L is upper triangular.

Then, taking into account the rule of transposition of matrices, we can write

_.

As a result, we get

_{, , .}

Here the rows of the matrix are orthonormal.

Comparing

and

we get

_{, .}

Thus, we again obtain the orthonormal initial values of the unknown vector-valued functions of the solution.