Volume 18, Issue 5

To stabilize the solution of a nonlinear parabolic equation with prescribed rate by the boundary conditions we construct the discrete analogue of the algorithm proposed for the differential case by A. V. Fursikov. The informative part of the work is the analysis and use of the procedure for the approximate projection of the solution onto a specially constructed manifold. Qualitative differences between the discrete and continuous approaches are fixed.

To stabilize the solution of a nonlinear parabolic equation with prescribed rate by the boundary conditions we construct the discrete analogue of the algorithm proposed for the differential case by A. V. Fursikov. The informative part of the work is the analysis and use of the procedure for the approximate projection of the solution onto a specially constructed manifold. Qualitative differences between the discrete and continuous approaches are fixed.

In this paper, we investigate the convergence rate of the multigrid method for solving the Helmholtz equation with boundary Dirichlet conditions for a sequence of quasinested adaptive grids. The symmetrical Gauss–Seidel method is used as smoothing. There is made a comparison with the method based on the fast discrete Fourier transform in application scenarios. Earlier there was investigated the convergence of the multigrid method in which the smoothing Jacobi and Richardson operators were used. We give the results of numerical experiments that show the higher convergence rate of the algorithm proposed.

In this paper, we investigate the convergence rate of the multigrid method for solving the Helmholtz equation with boundary Dirichlet conditions for a sequence of quasinested adaptive grids. The symmetrical Gauss–Seidel method is used as smoothing. There is made a comparison with the method based on the fast discrete Fourier transform in application scenarios. Earlier there was investigated the convergence of the multigrid method in which the smoothing Jacobi and Richardson operators were used. We give the results of numerical experiments that show the higher convergence rate of the algorithm proposed.

In this paper, we consider the problem of numerical solution of the system of forward– backward stochastic differential equations and its Cauchy problem for a quasilinear parabolic equation. We propose a layer method of solving the Cauchy problem with simultaneous estimation of the solution gradient, which is based on the probabilistic representation of the solution. We consider the ways of constructing layer methods for nonlinear boundary value problems. The system of stochastic differential equations is solved by the Euler method using the preliminarily obtained numerical solution of a quasilinear equation and its gradient.

In this paper, we consider the problem of numerical solution of the system of forward– backward stochastic differential equations and its Cauchy problem for a quasilinear parabolic equation. We propose a layer method of solving the Cauchy problem with simultaneous estimation of the solution gradient, which is based on the probabilistic representation of the solution. We consider the ways of constructing layer methods for nonlinear boundary value problems. The system of stochastic differential equations is solved by the Euler method using the preliminarily obtained numerical solution of a quasilinear equation and its gradient.

We consider problems whose mathematical model is associated with a certain Markov chain, in this case it is necessary to estimate the linear functionals of the solution to the corresponding integral equation of the second kind. Auxiliary variables whose values define transitions in the initial chain are included in the number of coordinates of the phase space to construct the weight modifications of numerical statistical simulation. Auxiliary weight after the realization of each auxiliary random variable is multiplied by the ratio of the corresponding densities of the initial distribution to the modelled one. In this paper, we solve the minimization problem of the variances of the estimates for linear functionals by varying the modelled distribution density of one (the first in order) auxiliary, maybe vector, random variable.

We consider problems whose mathematical model is associated with a certain Markov chain, in this case it is necessary to estimate the linear functionals of the solution to the corresponding integral equation of the second kind. Auxiliary variables whose values define transitions in the initial chain are included in the number of coordinates of the phase space to construct the weight modifications of numerical statistical simulation. Auxiliary weight after the realization of each auxiliary random variable is multiplied by the ratio of the corresponding densities of the initial distribution to the modelled one. In this paper, we solve the minimization problem of the variances of the estimates for linear functionals by varying the modelled distribution density of one (the first in order) auxiliary, maybe vector, random variable.

We consider discrete approximations of the first initial boundary value problem for a singularly perturbed heat equation on a domain outside the 'inclusion', i.e. on the domain = R ×[ t 0 , T ] excluding a rectangle whose sides are noncollinear to the x-, t -axes. The solution of the problem has boundary moving layers, regular by their behaviour, in a neighbourhood of the outside part of the boundary and parabolic transition layers in a neighbourhood of characteristics tangent to the inclusion. In addition, the solution of the problem has a discontinuity-type singularity occurring at the instant the inclusion appears. Using the fitted operator method (in a neighbourhood of the layer appearance, where the singularity develops) and the refining mesh technique (in a neighbourhood of the layers), we construct a special difference scheme convergent ε-uniformly almost everywhere in the grid domain except for the set of the solution discontinuity. When constructing the scheme we use the new coordinates in which the location of boundary layers becomes stationary.

We consider discrete approximations of the first initial boundary value problem for a singularly perturbed heat equation on a domain outside the 'inclusion', i.e. on the domain = R ×[ t 0 , T ] excluding a rectangle whose sides are noncollinear to the x-, t -axes. The solution of the problem has boundary moving layers, regular by their behaviour, in a neighbourhood of the outside part of the boundary and parabolic transition layers in a neighbourhood of characteristics tangent to the inclusion. In addition, the solution of the problem has a discontinuity-type singularity occurring at the instant the inclusion appears. Using the fitted operator method (in a neighbourhood of the layer appearance, where the singularity develops) and the refining mesh technique (in a neighbourhood of the layers), we construct a special difference scheme convergent ε-uniformly almost everywhere in the grid domain except for the set of the solution discontinuity. When constructing the scheme we use the new coordinates in which the location of boundary layers becomes stationary.