Petr N. Vabishchevich

Numerical simulation of convection/diffusion phase change problems : a review

A review of numerical techniques for the solution of heat and mass transfer problems with solid/liquid phase change is presented. The mathematical model for a description of a thermal field is based on the conventional Stefan approximation for the evaluation of phase change and the Navier-Stokes equations in the Boussinesq approximation for convective flows of a melt. Two basic approaches for the solution of these problems with a free boundary (phase change interface) are considered. The first approach is connected with interface-fitting algorithms (referred to in the work as variable grid methods), the second one with interface-smearing (fixed grid) methods. Fixed grid methods for the investigation of hydrodynamical phenomena in a varying calculation domain are constructed using various modifications of a penalty method.

A finite difference analysis of Biot's consolidation model

In this paper, stability estimates and convergence analysis of finite difference methods for the Biots consolidation model are presented. Initially central differences for space discretization and a weighed two-level time scheme are analyzed. To improve some stability and convergence limitations for this scheme we also consider space discretizations on MAC type grids (staggered grids). Numerical results are given to illustrate the obtained theoretical results.

A numerical study on natural convection of a heat-generating fluid in rectangular enclosures

Abstract Unsteady natural convection of a heat-generating fluid in the enclosures of a rectangular section with isothermal or adiabatic rigid walls is investigated numerically in the present work. Using the high-performance finite-difference scheme in the 2D stream function-vorticity formulation, developed by the authors, the peculiarities of convective heat transfer are studied in a wide range of thermal and geometric parameters for the laminar regime of fluid motion. Steady-state as well as oscillating solutions obtained in this work are compared with available numerical and experimental results of other researchers.

Numerically solving an equation for fractional powers of elliptic operators

An equation for a fractional power of the second-order elliptic operator is considered. It is solved numerically using a time-dependent problem for a pseudo-parabolic equation. For the auxiliary Cauchy problem, the standard two-level schemes are applied. Stability conditions are obtained for the fully discrete schemes under the consideration. The numerical results are presented for a model of two-dimensional problem with a fractional power of an elliptic operator. The dependence of accuracy on grids in time and in space is studied.

Splitting schemes for poroelasticity and thermoelasticity problems

Abstract In this work, we consider the coupled systems of linear unsteady partial differential equations, which arise in the modelling of poroelasticity processes. Stability estimates of weighted difference schemes for the coupled system of equations are presented. Approximation in space is based on the finite element method. We construct splitting schemes and give some numerical comparisons for typical poroelasticity problems. The results of numerical simulation of a 3D problem are presented. Special attention is given to using high performance computing systems.

A Substructuring Domain Decomposition Scheme for Unsteady Problems

Abstract Domain decomposition methods are used for the approximate solution of boundary-value problems for partial differential equations on parallel computing systems. Specific features of unsteady problems are fully taken into account in iteration-free domain decomposition schemes. Regionally-additive schemes are based on various classes of splitting schemes. In this paper we highlight a class of domain decomposition schemes which are based on the partition of the initial domain into subdomains with common boundary nodes. Using a partition of unity we construct and analyze unconditionally stable schemes for domain decomposition based on a two-component splitting: the problem within each subdomain and the problem at their boundaries. As an example we consider a Cauchy problem of first or second order in time with a non-negative self-adjoint second order operator in space. The theoretical discussion is supplemented with the numerical solution of a model problem for a two-dimensional parabolic equation.

Two-level finite difference scheme of improved accuracy order for time-dependent problems of mathematical physics

In the theory of finite difference schemes, the most complete results concerning the accuracy of approximate solutions are obtained for two- and three-level finite difference schemes that converge with the first and second order with respect to time. When the Cauchy problem is numerically solved for a system of ordinary differential equations, higher order methods are often used. Using a model problem for a parabolic equation as an example, general requirements for the selection of the finite difference approximation with respect to time are discussed. In addition to the unconditional stability requirements, extra performance criteria for finite difference schemes are presented and the concept of SM stability is introduced. Issues concerning the computational implementation of schemes having higher approximation orders are discussed. From the general point of view, various classes of finite difference schemes for time-dependent problems of mathematical physics are analyzed.

Computational identification of the right-hand side of a parabolic equation

Among inverse problems for partial differential equations, a task of interest is to study coefficient inverse problems related to identifying the right-hand side of an equation with the use of additional information. In the case of nonstationary problems, finding the dependence of the right-hand side on time and the dependence of the right-hand side on spatial variables can be treated as independent tasks. These inverse problems are linear, which considerably simplifies their study. The time dependence of the right-hand side of a multidimensional parabolic equation is determined using an additional solution value at a point of the computational domain. The inverse problem for a model equation in a rectangle is solved numerically using standard spatial difference approximations. The numerical algorithm relies on a special decomposition of the solution whereby the transition to a new time level is implemented by solving two standard grid elliptic problems. Numerical results are presented.

Interpolation finite difference schemes on grids locally refined in time

Numerical methods in which the mesh is locally refined are widely used for problems with singularities in the solution. In this case, approaches with refining the grid in both space and time are being developed. In this paper, we consider a class of finite difference schemes with local refinement of the grid in time to solve the problems numerically; here we compute the numerical solution on a finer time grid in a part of the domain. We consider a model Dirichlet problem for a second-order parabolic equation on a rectangle. We analyze the accuracy of completely implicit schemes with the simplest interpolated interface conditions on the boundary of the adaptation domain. On the basis of the maximum principle, the unconditional convergence of these schemes in the uniform norm is shown, and the rate of convergence is analyzed.

Operator-splitting schemes for the stream function-vorticity formulation

Abstract The paper deals with new implicit finite-difference schemes for solving the time-dependent incompressible Navier-Stokes equations in the stream function-vorticity formulation. New skew-symmetric second-order approximations are developed for the convective terms. The fully implicit implementation of both (no-slip and no-permeability) boundary conditions is constructed on the basis of the operator-splitting technique. For the discrete solution, there does exist an a priori estimate of the kinetic energy integral, free of any restrictions on grid parameters. The estimate is similar to the corresponding one for the solution of the differential problem and guarantees boundedness of the solution. To validate the new algorithms, a lid-driven cavity flow of an incompressible fluid has been considered for a range of Reynolds numbers.