Home – Home – Jornals – Numerical Analysis and Applications 2016 number 2

In this paper, two new algorithms for the simulation of homogeneous random fields are proposed. Both algorithms are based on the widespread algorithm “in rows and columns” for the simulation of the Gaussian fields with special correlation functions. Applying the algorithms developed makes possible to efficiently simulate homogeneous random fields with non-convex correlation functions.

A mixed problem for a one-dimensional heat equation with several versions of initial and boundary conditions is considered. Explicit and implicit schemes are applied for the solution. The sweep method and the iteration methods are used for the implicit scheme for solving the implicit system of equations. The numerical filtration of a finite sequence of results obtained for different grids with an increasing number of nodal points is used to analyze errors of the method and rounding. In addition, to investigate the rounding errors, the results obtained with several lengths of the machine word mantissa are compared. The numerical solution of the mixed problem for the wave equation is studied by similar methods. The occurrence of deterministic dependencies of the error in the numerical method and the rounding on spatial coordinates, time and the number of nodes is revealed. The source models to describe the behavior of errors in terms of time are based on the analysis of the results of numerical experiments for different versions of conditions of problems. In accord with such models, which were verified by the experiment, the errors can increase, decrease or stabilize depending on conditions over time similar to changing the energy or mass.

A numerical algorithm for computing tsunami wave front amplitude is proposed. The first step consists in solving an appropriate eikonal equation. The eikonal equation is solved by the Godunov approach and the bicharacteristic method. The qualitative comparison of the two above methods is described. Then a change in variables associated with the eikonal solution is introduced. At the last step, using the expansion of the fundamental solution of shallow water equations in the sum of singular and regular parts, we obtain the Cauchy problem for the wave amplitude. This approach allows one to reduce computer costs. The numerical results are presented.

In this paper, we present new interesting fourth-order optimal families of Chebyshev-Halley type methods free from second-order derivatives. In terms of computational cost, each member of the families requires two functions and one first-order derivative evaluation per iteration, so that their efficiency indices are 1.587. It is found by way of illustration that the proposed methods are useful in high precision computing environment. Moreover, it is also observed that larger basins of attraction belong to our methods, whereas the other methods are slow and have darker basins, while some of the methods are too sensitive to the choice of the initial guess.

A new approach to the decomposition method of a three-dimensional computational domain into subdomains, adjoint without overlapping, which is based on a direct approximation of the Poincare-Steklov equation at the conjugation interface, is proposed. With the use of this approach, parallel algorithms and technologies for three-dimensional boundary value problems on quasi-structured grids are presented. The experimental evaluation of the parallelization efficiency on the solution of the model problem on quasi-structured parallelepipedal coordinated and uncoordinated grids is given.

An approach to optimization of trading strategies (algorithms) based on indicators of financial markets and evolutionary computation is described. A new version of the differential evolution algorithm for the search for optimal parameters of trading strategies for the trading profit maximization is used. The experimental results show that this approach can considerably improve the profitability of the trading strategies.

In the present paper, approximate analytical and numerical solutions to nonlinear eigenvalue problems arising in the nonlinear fracture mechanics in analysis of stress-strain fields near a crack tip under a mixed mode loading are presented. Asymptotic solutions are obtained by the perturbation method (the small artificial parameter method). The artificial small parameter is a difference between the eigenvalue corresponding to the nonlinear eigenvalue problem and the eigenvalue related to the linear “undisturbed” problem. It is shown that the perturbation technique gives an effective method of solving nonlinear eigenvalue problems in the nonlinear fracture mechanics. Comparison of numerical and asymptotic results for different values of the mixity parameter and hardening exponent shows good agreement. Thus, the perturbation theory technique for studying nonlinear eigenvalue problems is offered and applied to eigenvalue problems arising from the fracture mechanics analysis in the case of a mixed mode loading.

We study the grid-characteristic methods for solving hyperbolic systems using a high order interpolation on unstructured tetrahedral and triangular grids for approximation. We consider the interpolation with orders from the first to the fifth included. Also, one-dimensional finite difference schemes appropriate for the considered methods are given. We study these schemes in terms of stability. The grid-characteristic method on unstructured triangular and tetrahedral grids are successfully used for solving the seismic prospecting problems, including, seismic prospecting in the conditions of the Arctic shelf and permafrost, as well as for solving seismic problems, problems of dynamic deformation and destruction, studying anisotropic composite materials.