Vyacheslav G. Tsybulin

Finite-difference approximations and cosymmetry conservation in filtration convection problem

Abstract We consider finite-difference approximations of the planar Darcy convection problem and study the effect of different discretizations with respect to preservation of cosymmetry. The important feature of cosymmetrical systems is the existence of the family of stationary regimes with a spectrum that varies over a family, and an accurate computation of the family of equilibria is the key point of our consideration. Different approximations of Jacobians are compared and we found that the Arakawa scheme provides the most accurate results due to its conservation properties. Some evidence of family degeneration is presented when an inappropriate approximation was used.

Mimetic discretization of two dimensional Darcy convection

Abstract We consider discretization of the planar convection of the incompressible fluid in a porous medium filling rectangular enclosure. This problem belongs to the class of cosymmetric systems and admits an existence of a continuous family of steady states in the phase space. Mimetic finite-difference schemes for the primitive variables equation are developed. The connection of a derived staggered discretization with a finite-difference approach based on the stream function and temperature equations is established. Computations of continuous cosymmetric families of steady states are presented for the case of uniform and nonuniform grids.

Conservative Finite Difference Schemes for Cosymmetric Systems

We consider the application of computer algebra for the derivation of the formula for the preservation of the cosymmetry property through discretization of partial differential equations. The finite difference approximations of differential operators for both regular and staggered grids are derived and applied to the planar filtration-convection problem.

Calculation of Families of Stationary Filtration Convection Regimes in a Narrow Container

A plane Darcy filtration convection problem for rectangular containers elongated in the vertical direction is considered. By the spectral‐difference method, which preserved cosymmetry of the initial problem, evolution of families of stationary regimes from the onset of instability on the primary family till the collision of the families is studied.

DYNAMICS OF NUMERICAL METHODS FOR COSYMMETRIC ORDINARY DIFFERENTIAL EQUATIONS

The dynamics of numerical approximation of cosymmetric ordinary differential equations with a continuous family of equilibria is investigated. Nonconservative and Hamiltonian model systems in two dimensions are considered and these systems are integrated with several first-order Runge–Kutta methods. The preservation of symmetry and cosymmetry, the stability of equilibrium points, spurious solutions and transition to chaos are investigated by presenting analytical and numerical results. The overall performance of the methods for different parameters is discussed.

Staggered grids discretization in three-dimensional Darcy convection

We consider three-dimensional convection of an incompressible fluid saturated in a parallelepiped with a porous medium. A mimetic finite-difference scheme for the Darcy convection problem in the primitive variables is developed. It consists of staggered nonuniform grids with five types of nodes, differencing and averaging operators on a two-nodes stencil. The nonlinear terms are approximated using special schemes. Two problems with different boundary conditions are considered to study scenarios of instability of the state of rest. Branching off of a continuous family of steady states was detected for the problem with zero heat fluxes on two opposite lateral planes.

Invariant sets and attractors of quadratic mapping of plane: Computer experiment and analytical treatment

Consider a dynamical system generated by a quadratic mapping of the plane with parameters . This 2-parameter family arises as a finite -difference approximation of an ODE system with special quadratic nonlinearity and also as a leading part of a map on a center manifold (in some problems of intersections of bifurcations). Computer experiments on attractors with varyingλand μgives rise to finding and deriving some analytical results. We determine several interesting ivariant mainfolds and sets (invariant straight lines,invariant crosses (line-line and line -parabola)and circle) that exist for definite values of parameters. Next we point out the invariant measure (density)on the invariant disc bounded by the invariant circle.

Natural convection in porous annular domains

Natural convection of the incompressible fluid in the porous media based on the Darcy hypothesis (Lapwood convection) gives an intriguing branching off of one-parameter family of steady patterns. This scenario may be suppressed in computations when governing equations are approximated by schemes which do not preserve the cosymmetry property. We consider the problem for the annular porous domain in polar coordinates and derive a mimetic finite-difference scheme. This scheme allows to compute the family of convective regimes accurately and to detect the instabilities on some parts of the family.

Staggered grids for three-dimensional convection of a multicomponent fluid in a porous medium

Convection in a porous medium may produce strong nonuniqueness of patterns. We study this phenomena for the case of a multicomponent fluid and develop a mimetic finite-difference scheme for the three-dimensional problem. Discretization of the Darcy equations in the primitive variables is based on staggered grids with five types of nodes and on a special approximation of nonlinear terms. This scheme is applied to the computer study of flows in a porous parallelepiped filled by a two-component fluid and with two adiabatic lateral planes. We found that the continuous family of steady stable states exists in the case of a rather thin enclosure. When the depth is increased, only isolated convective regimes may be stable. We demonstrate that the non-mimetic approximation of nonlinear terms leads to the destruction of the continuous family of steady states.

A Mimetic Finite-Difference Scheme for Convection of Multicomponent Fluid in a Porous Medium

A mimetic finite-difference scheme for the equations of three-dimensional convection of a multicomponent fluid in a porous medium is developed. The discretization is based on staggered grids with five types of nodes (velocities, pressure, temperature, and mass fractions) and on a special approximation of nonlinear terms. Computer experiments have revealed the continuous family of steady states in the case of the zero heat fluxes through two opposite lateral planes of parallelepiped.