Increase of the order of approximation of the Godunov scheme on the basis of the solution of a generalized Riemann problem.

*(Russian)*Zbl 0714.65080The finite-difference Godunov scheme is an explicit divergence scheme, which is frequently used for the examination of solid environment mechanic problems. The vector-function of parameters describing the state of the environment in this method in the given time is approximated by a piecewise constant function.

In the paper the generalized problem of decomposition at the arbitrary point of discontinuity is solved in the situation where in the initial time point on the left and on the right side of the discontinuity point, the decomposition of the dynamical parameters is not constant. In some neighbourhood of the point of discontinuity the analytical solution of this problem is obtained and is in addition used for the construction of the Godunov scheme of second order of approximation.

In the paper the generalized problem of decomposition at the arbitrary point of discontinuity is solved in the situation where in the initial time point on the left and on the right side of the discontinuity point, the decomposition of the dynamical parameters is not constant. In some neighbourhood of the point of discontinuity the analytical solution of this problem is obtained and is in addition used for the construction of the Godunov scheme of second order of approximation.

Reviewer: J.Vaníček

##### MSC:

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

65M30 | Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs |