Home – Home – Jornals – Numerical Analysis and Applications 2020 number 3

An ensemble of independent numerical solutions enables one to construct a hypersphere around the approximate solution that contains the true solution. The analysis is based on some geometry considerations, such as the triangle inequality and the measure concentration in the spaces of large dimensions. As a result, there appears the feasibility for non-intrusive postprocessing that provides the error estimation on the ensemble of solutions. The numerical tests for two-dimensional compressible Euler equations are provided that demonstrates properties of such postprocessing.

New h-, p- and hp-versions of the least-squares collocation method are proposed and implemented for solving the Dirichlet problem for the Poisson equation. The paper considers some examples of solving problems with singularities such as large gradients, high growth rate of solution derivatives with increasing the order of differentiation, discontinuity of the second-order derivatives at the angular points of the domain boundary, and the oscillating solution with different frequencies in the presence of an infinite discontinuity for derivatives of any order. The new versions of the method are based on a special selection of collocation points in the roots of the Chebyshev polynomials of the first kind. Basis functions are defined as a product of the Chebyshev polynomials. The behavior of the numerical solution on a sequence of grids and with an increase in the degree of the approximating polynomial has been analyzed using exact analytical solutions. The formulas for the continuation operation necessary for the transition from a coarse mesh to a finer one on a multi-grid complex in the Fedorenko method have been obtained.

This paper deals with the to boundary value problems for pseudoparabolic equations of fractional order with the Bessel operator with variable coefficients with non-local boundary conditions of the integral type and difference methods for their solutions. To solve the considered problems a priori estimates in differential and difference interpretations are obtained, which means the uniqueness and stability of solutions by initial data and the right-hand side, as well as the convergence of the solution of the difference problem to the solution of the corresponding differential problem.

Analytical and numerical methods for solving inverse problems of logarithmic and the Newtonian potentials are investigated. The following contact problem in the case of a Newtonian potential is considered. In the domain Ω{Ω: -l ≤ x,y ≤ l, H υ(x,y) ≤ z ≤ H}, sources with the density ρ(x,y), perturbing the Earth’s gravitational field, are distributed. Here, υ(x,y) is a non-negative finite function with the support Ω = [l,l]², 0 ≤ υ(x,y) ≤ H. It is required to simultaneously restore the depth H of the occurrence of the contact surface z = H, the density ρ(x,y) of sources, and the function υ(x,y). The methods of simultaneous determination are based on the use of nonlinear models of potential theory which are developed in the paper. The following kinds of information are used as the basic ones: 1) values of the gravity field and its first and second derivatives; 2) values of the gravity field at the different heights. The possibility of the simultaneous recovery of the functions ρ( x,y), υ(x,y) and the constants H in the analytical form is demonstrated. Iterative methods for their simultaneous recovery. The model examples demonstrate the effectiveness of the proposed numerical methods are constructed.

For one of cubic equations of state, the Redlich-Kwong, the inverse curve, which characterizes a change in a sign of the Joule-Thomson coefficient, has been constructed. The range of reduced pressure and temperature corresponds to the values intrinsic of technological changes in production and transport systems of natural gas.

In this paper, an operator iterative procedure for constructing of the orthogonal projection of a vector on a given subspace is proposed. The algorithm is based on the Euclidean ortogonalization of power sequences of a special linear transformation generated by the original subspace. For consistent systems of linear algebraic equations, a numerical method based on this idea is proposed. Numerical results are presented.

The shallow gas in the ground geological layers of the water space is of great danger for the drilling rigs in the case of an accident opening of the gas deposits. Gas starts rising towards the surface of water, and sooner or later, the gas emission into the atmosphere threatens the environment. It is very important to be able to forecast the gas emissions in order to prevent the catastrophic consequences with the destruction of drilling rigs and people fatalities. This paper presents the results for the numerical modeling of seismic waves propagation in models with gas deposits through the layered soil towards the surface of water for the 3D case. The modeling was carried out for the 4-year period for the layers, which are located at the depth of 1000 m from the bottom of the sea. The results of the computations (the wave pictures and seismograms) show the approach of gas to the surface of water for the 4th year of the computations. The consistency of the results for the 3D problem with the results for the 2D problem, early obtained by the authors, is very important for the further research into the area in question.

We investigate the influence of the Wiener and the Poisson random noises on the behavior of the linear and Van der Pol oscillators with the help of the statistical modeling method. For a linear oscillator, the analytical expression of the autocovariance function of the solution to stochastic differential equation (SDE) is obtained. This expression along with the formulas of mathematical expectation and variance of the SDE solution allows us to carry out the parametric analysis and to investigate the accuracy of estimates of moments of the numerical solution to the SDE obtained with the help of the generalized Euler explicit method. For the Van der Pol oscillator, the influence of the Poisson component on the oscillation nature of the first and the second moments of the SDE solution with a large value of jumps is numerically investigated.