Home – Home – Jornals – Numerical Analysis and Applications 2023 number 4

A new iterative method for the modeling of non-Gaussian random vectors with given marginal distributions and covariance matrix is proposed in this paper. The algorithm is compared with another iterative algorithm for the modeling of non-Gaussian vectors, which is based on reordering a sample of independent random variables with given marginal distributions. Our numerical studies show that both algorithms are equivalent in terms of the accuracy of reproducing the given covariance matrix, but the proposed algorithm turns out to be more efficient in terms of memory usage and, in many cases, is faster than the other one.

In this paper, iterative stochastic simulation algorithms for the Lamè equation describing the displacements of an isotropic elastic body are constructed. Three different stochastic methods are proposed: the first one is based on a global algorithm of random walk on spheres to compute the solution and its derivatives for an anisotropic diffusion equation. It does not use grids and does not require large amounts of RAM. The second method is based on a randomized algorithm for solving large systems of linear equations and requires the introduction of a grid. The third method is also grid-based and uses a random walk algorithm. All three methods implement an iterative process, at each step of which anisotropic diffusion equations are solved. The paper provides a comparative analysis of the proposed methods and discusses the limits of applicability of each of them.

A review of the relatively little-known matrix class, called coninvolutions, is given. The properties of these matrices are compared with those of the well studied involutory matrices or, briefly, involutions.

The Lax-Friedrichs scheme is traditionally considered an alternative to the Godunov scheme, since it does not require solving the Riemann problem. In the equations of special relativistic hydrodynamics, the speed of light is a natural limitation of the wave propagation speed. The use of such an upper estimate of the slopes of characteristics in the schemes of Roe, the Rusanov type, or the Harten-Lax-Van Leer family leads to a construction equivalent to the Lax-Friedrichs scheme. Due to the absolute robustness of the scheme, a number of software implementations have been developed on its basis for modeling relativistic gas flows. In this paper, we propose a piecewise parabolic reconstruction of the physical variables to reduce dissipation of the numerical method. The use of such a reconstruction in the Lax-Friedrichs scheme allows us to obtain an absolutely robust simple scheme of high-order accuracy on smooth solutions and with small dissipation at the discontinuities. The computational experiments carried out in the article confirm these properties of the scheme.

A new correlative-grid approximation of a homogeneous and isotropic random density field is introduced for the effective numerically-analytical investigation of overexponential growth of the mean particles flux in a random medium with multiplication. In this case the complexity of the particle trajectory realization is not dependent on the correlation scale. For the correlative-grid approximation the possibility of a Gaussian asymptotics of the mean particles multiplication rate is justified for a random field of bounded density. It ensures a superexponential growth of the flux in some initial time interval. An estimate of further overexponential flux growth is constructed based on some test computations.

A process of searching on the sphere for the best (in a sense) cubature formulas that are invariant under the transformations of the icosahedral rotation group is described. The parameters of the best cubature formulas of this symmetry type up to the 30th order of accuracy are given to 16 significant digits. A table which contains the main characteristics of all the best to date cubature formulas of the icosahedral rotation group up to the 79th order of accuracy is given.

To calculate flows of a gas-liquid mixture, a modified inverse method of characteristics is proposed, in whose algorithm an additional fractional time step is introduced, which makes it possible to carry out calculations with a large time step without loss of accuracy and stability. A formulation of boundary conditions on curvilinear walls is discussed in relation to the multidimensional nodal method of characteristics, which is based on splitting along the coordinate directions of the original system of equations into a number of one-dimensional subsystems. For the boundary points located on curvilinear impenetrable surfaces, a calculation method based on the method of fictitious nodes is proposed. When testing the modified method, the supersonic interaction of a homogeneous dispersed flow with a barrier is calculated for a flow regime with an attached shock wave. Problems of steady mixture flows near an external obtuse angle, as well as near a cone, which are analogues of the Prandtl-Meyer and Busemann flows in gas dynamics, are solved. The calculation results are compared with available self-similar solutions, and a satisfactory agreement is reached.

A difference scheme of the predictor-corrector type is constructed for solving nonlinear dispersion equations of wave hydrodynamics with a high order of approximation of the dispersion relation, based on splitting of the original system of equations into a hyperbolic system and a scalar equation of the elliptic type. A dissipation and dispersion analysis of the new scheme is performed, a condition for its stability is obtained, and a formula for the phase error is written and analyzed. Parameters are found at which the phase characteristics of the difference scheme, the nonlinear-dispersive model approximated by it, and the full model of potential flows have the same order of accuracy.