arXiv subjects

Petr N. Vabishchevich

Publications and source records attributed to Petr N. Vabishchevich.

At least 37 records · Page 2Linked to original sources

arXiv2019

Computational identification of adsorption and desorption parameters for pore scale transport in periodic porous media

Computational identification of unknown adsorption and desorption rates is discussed in conjunction with reactive flow considered at pore scale. The reactive transport is governed by incompressible Stokes equations, coupled with convection-diffusion equation for species transport. The surface reactions, namely adsorption and desorption, are accounted via Robin boundary conditions. Henry and Langmuir isotherms are considered. Measured concentration of the specie at the outlet of the domain has to be provided to carry out the identification procedure. Deterministic and stochastic parameter identification approaches are considered. The influence of the noise in the measurements on the accuracy of the identified parameters is discussed. Multistage identification procedure is suggested for the considered class of problems. The proposed identification approach is applicable for different geometries (random and periodic) and for a range of process parameters. In this paper the potential of the approach is demonstrated in identifying parameters of Langmuir isotherm for low Peclet and low Damkoler numbers reactive flow in a 2D periodic porous media with circular inclusions. Simulation results for random porous media and other regime parameters are subject of follow up papers. Finite element approximation in space and implicit time discretization are exploited.

math.NA↗

arXiv2019

Approximate representation of the solutions of fractional elliptical BVP through the solution of parabolic IVP

Boundary value problem for a fractional power of an elliptic operator is considered. An integral representation by means of a standard solution problem for parabolic equations is used to solve such problems. Quadrature generalized Gauss-Laguerre formulas are constructed. We examine the effect of key parameters on the accuracy of the approximate solution: the number of nodes of the quadrature and fractional power of the operator. Computational experiments were performed to model two-dimensional problem with a fractional power of an elliptic operator.

math.NA↗

arXiv2019

Approximation of a fractional power of an elliptic operator

Some mathematical models of applied problems lead to the need of solving boundary value problems with a fractional power of an elliptic operator. In a number of works, approximations of such a nonlocal operator are constructed on the basis of an integral representation with a singular integrand. In the present paper, new and more convenient integral representations are proposed for operators with fractional powers. Approximations are based on the classical quadrature formulas. The results of numerical experiments on the accuracy of quadrature formulas are presented. The proposed approximations are used for numerical solving a model two-dimensional boundary value problem for fractional diffusion.

math.NA↗

arXiv2019

Numerical modeling of neutron transport in SP3 approximation by finite element method

The SP3 approximation of the neutron transport equation allows improving the accuracy for both static and transient simulations for reactor core analysis compared with the neutron diffusion theory. Besides, the SP3 calculation costs are much less than higher order transport methods (SN or PN). Another advantage of the SP3 approximation is a similar structure of equations that is used in the diffusion method. Therefore, there is no difficulty to implement the SP3 solution option to the multi-group neutron diffusion codes. In this work, the application of the SP3 methodology based on solution of the λ- and α-spectral problems has been tested for the IAEA-2D and HWR reactor benchmark tests. The FEM is chosen to achieve the 3D geometrical generality, using GMSH as a generic mesh generator. The results calculated with the diffusion and SP3 methods are compared with the reference transport calculation results. It was found for the HWR reactor test that some eigenvalues are complex when calculating using both diffusion and SP3 options.

cs.CE↗

arXiv2018

Existence of a system of discrete volume-localized quantum levels for charged fullerenes

In the framework of a simple physical model, we demonstrate the existence of a system of discrete short-lifetime quantum levels for electrons in the potential well of the self-consistent field of charged fullerenes and onion-like structures. For electrons, in the case of positively charged fullerenes and onion-like structures, combining analytic and numeric considerations we find that the energy of the volume-localized levels ranges from 1 eV to 100 eV.

physics.atm-clus↗

arXiv2018

Computational identification of the lowest space-wise dependent coefficient of a parabolic equation

In the present work, we consider a nonlinear inverse problem of identifying the lowest coefficient of a parabolic equation. The desired coefficient depends on spatial variables only. Additional information about the solution is given at the final time moment, i.e., we consider the final redefinition. An iterative process is used to evaluate the lowest coefficient, where at each iteration we solve the standard initial-boundary value problem for the parabolic equation. On the basis of the maximum principle for the solution of the differential problem, the monotonicity of the iterative process is established along with the fact that the coefficient approaches from above. The possibilities of the proposed computational algorithm are illustrated by numerical examples for a model two-dimensional problem.

math.NA↗

arXiv2018

Numerical solution of boundary value problems for the eikonal equation in an anisotropic medium

A Dirichlet problem is considered for the eikonal equation in an anisotropic medium. The nonlinear boundary value problem (BVP) formulated in the present work is the limit of the diffusion-reaction problem with a reaction parameter tending to infinity. To solve numerically the singularly perturbed diffusion-reaction problem, monotone approximations are employed. Numerical examples are presented for a two-dimensional BVP for the eikonal equation in an anisotropic medium. The standard piecewise-linear finite-element approximation in space is used in computations.

math.NA↗

arXiv2018

Decoupling schemes for predicting compressible fluid flows

Numerical simulation of compressible fluid flows is performed using the Euler equations. They include the scalar advection equation for the density, the vector advection equation for the velocity and a given pressure dependence on the density. An approximate solution of an initial--boundary value problem is calculated using the finite element approximation in space. The fully implicit two-level scheme is used for discretization in time. Numerical implementation is based on Newton's method. The main attention is paid to fulfilling conservation laws for the mass and total mechanical energy for the discrete formulation. Two-level schemes of splitting by physical processes are employed for numerical solving problems of barotropic fluid flows. For a transition from one time level to the next one, an iterative process is used, where at each iteration the linearized scheme is implemented via solving individual problems for the density and velocity. Possibilities of the proposed schemes are illustrated by numerical results for a two--dimensional model problem with density perturbations.

cs.CE↗

arXiv2017

Two-level schemes for the advection equation

The advection equation is the basis for mathematical models of continuum mechanics. In the approximate solution of nonstationary problems it is necessary to inherit main properties of the conservatism and monotonicity of the solution. In this paper, the advection equation is written in the symmetric form, where the advection operator is the half-sum of advection operators in conservative (divergent) and non-conservative (characteristic) forms. The advection operator is skew-symmetric. Standard finite element approximations in space are used. The standart explicit two-level scheme for the advection equation is absolutly unstable. New conditionally stable regularized schemes are constructed, on the basis of the general theory of stability (well-posedness) of operator-difference schemes, the stability conditions of the explicit Lax-Wendroff scheme are established. Unconditionally stable and conservative schemes are implicit schemes of the second (Crank-Nicolson scheme) and fourth order. The conditionally stable implicit Lax-Wendroff scheme is constructed. The accuracy of the investigated explicit and implicit two-level schemes for an approximate solution of the advection equation is illustrated by the numerical results of a model two-dimensional problem.

math.NA↗

arXiv2017

Numerical simulation of oxidation processes in a cross-flow around tube bundles

An oxidation process is simulated for a bundle of metal tubes in a cross-flow. A fluid flow is governed by the incompressible Navier-Stokes equations. To describe the transport of oxygen, the corresponding convection-diffusion equation is applied. The key point of the model is related to the description of oxidation processes taking into account the growth of a thin oxide film in the quasi-stationary approximation. Mathematical modeling of oxidant transport in a tube bundle is carried out in the 2D approximation. The numerical algorithm employed in the work is based on the finite-element discretization in space and the fully implicit discretization in time. The tube rows of a bundle can be either in-line or staggered in the direction of the fluid flow velocity. The growth of the oxide film on tube walls is predicted for various bundle structures using the developed oxidation model.

cs.CE↗

arXiv2017

State change modal method for numerical simulation of dynamic processes in a nuclear reactor

Modeling of dynamic processes in nuclear reactors is carried out, mainly, on the basis of the multigroup diffusion approximation for the neutron flux. The basic model includes a multidimensional set of coupled parabolic equations and ordinary differential equations. Dynamic processes are modeled by a successive change of the reactor states, which are characterized by given coefficients of the equations. In the modal method, the approximate solution is represented as an expansion on the first eigenfunctions of some spectral problem. The numerical-analytical method is based on the use of dominant time-eigenvalues of a multigroup diffusion model taking into account delayed neutrons. Numerical simulations of the dynamic process were performed in the framework of the two-group approximation for the VVER-1000 reactor test model. The last is characterized by the fact that some eigenvalues are complex.

cs.CE↗

arXiv2017

Fundamental mode exact schemes for unsteady problems

The problem of increasing the accuracy of an approximate solution is considered for boundary value problems for parabolic equations. For ordinary differential equations (ODEs), nonstandard finite difference schemes are in common use for this problem. They are based on a modification of standard discretizations of time derivatives and, in some cases, allow to obtain the exact solution of problems. For multidimensional problems, we can consider the problem of increasing the accuracy only for the most important components of the approximate solution. In the present work, new unconditionally stable schemes for parabolic problems are constructed, which are exact for the fundamental mode. Such two-level schemes are designed via a modification of standard schemes with weights using Padé approximations. Numerical results obtained for a model problem demonstrate advantages of the proposed fundamental mode exact schemes.

math.NA↗

arXiv2017

Two-component domain decomposition scheme with overlapping subdomains for parabolic equations

An iteration-free method of domain decomposition is considered for approximate solving a boundary value problem for a second-order parabolic equation. A standard approach to constructing domain decomposition schemes is based on a partition of unity for the domain under the consideration. Here a new general approach is proposed for constructing domain decomposition schemes with overlapping subdomains based on indicator functions of subdomains. The basic peculiarity of this method is connected with a representation of the problem operator as the sum of two operators, which are constructed for two separate subdomains with the subtraction of the operator that is associated with the intersection of the subdomains. There is developed a two-component factorized scheme, which can be treated as a generalization of the standard Alternating Direction Implicit (ADI) schemes to the case of a special three-component splitting. There are obtained conditions for the unconditional stability of regionally additive schemes constructed using indicator functions of subdomains. Numerical results are presented for a model two-dimensional problem.

math.NA↗

arXiv2017

Numerical solution of time-dependent problems with fractional power elliptic operator

An unsteady problem is considered for a space-fractional equation in a bounded domain. A first-order evolutionary equation involves a fractional power of an elliptic operator of second order. Finite element approximation in space is employed. To construct approximation in time, standard two-level schemes are used. The approximate solution at a new time-level is obtained as a solution of a discrete problem with the fractional power of the elliptic operator. A Pade-type approximation is constructed on the basis of special quadrature formulas for an integral representation of the fractional power elliptic operator using explicit schemes. A similar approach is applied in the numerical implementation of implicit schemes. The results of numerical experiments are presented for a test two-dimensional problem.

math.NA↗

arXiv2016

Splitting schemes for unsteady problems involving the grad-div operator

In this paper we consider various splitting schemes for unsteady problems containing the grad-div operator. The fully implicit discretization of such problems would yield at each time step a linear problem that couples all components of the solution vector. In this paper we discuss various possibilities to decouple the equations for the different components that result in unconditionally stable schemes. If the spatial discretization uses Cartesian grids, the resulting schemes are Locally One Dimensional (LOD). The stability analysis of these schemes is based on the general stability theory of additive operator-difference schemes developed by Samarskii and his collaborators. The results of the theoretical analysis are illustrated on a 2D numerical example with a smooth manufactured solution.

math.NA↗

arXiv2016

A Singularly Perturbed Boundary Value Problems with Fractional Powers of Elliptic Operators

A boundary value problem for a fractional power $0 < \varepsilon < 1$ of the second-order elliptic operator is considered. The boundary value problem is singularly perturbed when $\varepsilon \rightarrow 0$. It is solved numerically using a time-dependent problem for a pseudo-parabolic equation. For the auxiliary Cauchy problem, the standard two-level schemes with weights are applied. The numerical results are presented for a model two-dimen\-sional boundary value problem with a fractional power of an elliptic operator. Our work focuses on the solution of the boundary value problem with $0 < \varepsilon \ll 1$.

math.NA↗

arXiv2016

Iterative computational identification of a spacewise dependent the source in a parabolic equations

Coefficient inverse problems related to identifying the right-hand side of an equation with use of additional information is of interest among inverse problems for partial differential equations. When considering non-stationary problems, tasks of recovering the dependence of the right-hand side on time and spatial variables can be treated as independent. These tasks relate to a class of linear inverse problems, which sufficiently simplifies their study. This work is devoted to a finding the dependence of right-hand side of multidimensional parabolic equation on spatial variables using additional observations of the solution at the final point of time - the final overdetermination. More general problems are associated with some integral observation of the solution on time - the integral overdetermination. The first method of numerical solution of inverse problems is based on iterative solution of boundary value problem for time derivative with non-local acceleration. The second method is based on the known approach with iterative refinement of desired dependence of the right-hand side on spacial variables. Capabilities of proposed methods are illustrated by numerical examples for model two-dimensional problem of identifying the right-hand side of a parabolic equation. The standard finite-element approximation on space is used, the time discretization is based on fully implicit two-level schemes.

math.NA↗

arXiv2015

Numerical solving unsteady space-fractional problems with the square root of an elliptic operator

An unsteady problem is considered for a space-fractional equation in a bounded domain. A first-order evolutionary equation involves the square root of an elliptic operator of second order. Finite element approximation in space is employed. To construct approximation in time, regularized two-level schemes are used. The numerical implementation is based on solving the equation with the square root of the elliptic operator using an auxiliary Cauchy problem for a pseudo-parabolic equation. The scheme of the second-order accuracy in time is based on a regularization of the three-level explicit Adams scheme. More general problems for the equation with convective terms are considered, too. The results of numerical experiments are presented for a model two-dimensional problem.

math.NA↗