Volume 26, Issue 5

Numerical modelling of a series of problems with free boundaries is considered using the ISPH method in the presence of break modes. The main attention is paid to the calculation of the pressure field and determination of hydrodynamic loads on the vertical walls of the basin. The reliability of the results presented here is confirmed by the comparison with the results of other authors.

The method of solution of a partial spectral problem is constructed with the use of a special tensor structure of eigenvalues: in the case where the eigenvalues of a symmetric matrix represented in the QTT-format can be approximated in the form of a QTT-decomposition with a small number of parameters the method determines such approximate decomposition. The working time of the algorithm and the required memory are proportional to the logarithm of the total number of unknowns, whereas generally this dependence has been linear in the best case. It is shown how the efficiency of the tensor representation can be improved due to the use of a special block extension of the QTT-format. Convergence rate estimates are obtained for the modified method. The tensor method has been implemented in the block QTT-format on the base of the block conjugate gradient method. Numerical experiments have been performed for the solution of problems of mathematical physics in spaces of dimensions 2, 3, and more, and also for optimization problems for many-dimensional functions.

The problem of probing a medium with a ‘leader’ is considered in the paper. To do that, a biased modification of the ‘double local estimate’ of the Monte Carlo method is studied. In addition, its nonbiased modification with a finite variance (combined double local estimate) is proposed.

Monte Carlo methods for the solution of the nonlinear kinetic Boltzmann equation are considered. A special two-parametric weighted method is constructed. On the one hand, this method has improved performance, similar to the method of majorant frequency and, on the other hand, it is applicable to the case of an unbounded frequency of a pair collision.

This paper is focused on the development of numerical algorithms based on the natural element method for the solution of problems of incompressible fluid dynamics with free boundaries.