Chapter 2. Improvement of s.K.Godunov’s method of orthogonal sweep for solving boundary value problems with stiff ordinary differential equations.

2.1. The formula for the beginning of the calculation by s.K.Godunov’s sweep method.

Let us consider S.K.Godunov’s sweep method problem.

Suppose that we consider the shell of the rocket. This is a thin-walled tube. Then the system of linear ordinary differential equations will be of the 8th order, the matrix of coefficients will have the dimension 8x8, the required vector-function will have the dimension 8x1, and the matrices of the boundary conditions will be rectangular horizontal dimensions 4x8.

Then in S.K.Godunov’s method for such a problem the solution is sought in the following form [Godunov]:

,

or it can be written in the matrix form:

,

where vectors are linearly independent vector-solutions of the homogeneous system of differential equations, and the vector is a vector of a particular solution of the inhomogeneous system of differential equations.

Here is the matrix of dimension 8x4, and is the corresponding vector of dimension 4x1 with the required constants .

But in general, the solution for such a boundary-value problem with dimension 8 (outside the framework of S.K.Godunov's method) can consist not of 4 linearly independent vectors , but entirely of all 8 linearly independent solution vectors of the homogeneous system of differential equations:

And just the difficulty and problem of S.K.Godunov’s method is that the solution is sought with only half the possible vectors and constants, and the problem is that such a solution with half the constants must satisfy the conditions on the left edge (the starting edge for the sweep) for all possible values of the constants, in order to find these constants from the conditions on the right edge.

That is, in S.K.Godunov’s method, there is a problem of finding such initial values of the vectors , so that you can start the run from the left edge = 0, that is, that the conditions on the left edge are satisfied for any values of the constants .

Usually this difficulty is "overcome" by the fact that differential equations are written not through functionals, but through physical parameters and consider the simplest conditions on the simplest physical parameters so that the initial values can be guessed. That is, problems with complex boundary conditions can not be solved in this way: for example, problems with elastic conditions at the edges.

Below we propose a formula for the initiation of computations by S.K.Godunov’s method.

We perform the line orthonormalization of the matrix equation of the boundary conditions on the left edge:

,

where the matrix is rectangular and horizontal dimension 4x8.

As a result, we obtain an equivalent equation of boundary conditions on the left edge, but already with a rectangular horizontal matrix of dimension 4x8, which will have 4 orthonormal rows:

,

where, as a result of orthonormalization of the matrix , the vector is transformed into the vector .

How to perform line orthonormation of systems of linear algebraic equations can be found in [Berezin, Zhidkov].

We complete the rectangular horizontal matrix to a square non-degenerate matrix :

,

where a matrix of dimension 4х8 must complete the matrix to a non-degenerate square matrix of dimension 8х8.

As matrix rows, we can take those boundary conditions, that is, expressions of those physical parameters that do not enter the parameters of the left edge or are linearly independent with them. This is quite possible, since for boundary value problems there are as many linearly independent physical parameters as the dimensionality of the problem, that is, in this case there are 8 of them, and if 4 are given on the left edge, then 4 can be taken from the right edge.

We complete the orthonormalization of the constructed matrix , that is, we perform the line orthonormalization and obtain a matrix of dimension 8x8 with orthonormal rows:

.

We can write down that

.

Then, substituting in the formula of S.K. Godunov’s method, we get:

or

.

We substitute this last formula into the boundary conditions and obtain:

.

From this, we obtain that on the left-hand side the constants no longer influence anything, since and it remains only to find the vector from the expression:

.

But the matrix has a dimension of 4x8 and it must be supplemented to a square non-degenerate one in order to find the vector from the solution of the corresponding system of linear algebraic equations:

,

where is any vector, including a vector of zeros.

Hence we obtain by means of the inverse matrix:

.

Then the formula for starting the computation by S.K. Godunov's method is as follows:

.

From the theory of matrices [Gantmakher] it is known that if the matrix is orthonormal, then its inverse matrix is its transposed matrix. Then the last formula takes the form:

,

,

,

.

The column vectors of the matrix and the vertical convolution vector are linearly independent and satisfy the boundary condition .