Volume 2, Issue 4

In this paper, we consider a dead-oil model where the capillary pressure is neglected. In a multidimensional space and for a phase-by-phase upstream weigthing cell-centered finite volume scheme, we prove the pressure estimates, the existence of solutions to the discrete equations and the stability of the saturation calculation. This is done in the explicit case as well as in the implicit case. Some numerical tests show the convergence of the scheme.

In an attempt to show maximum-norm stability and smoothing estimates for finite element discretizations of parabolic problems on nonquasi-uniform triangulations we consider the lumped mass method with piecewise linear finite elements in one and two space dimensions. By an energy argument we derive resolvent estimate for the associated discrete Laplacian, which is then a finite difference operator on an irregular mesh, which show that this generates an analytic semigroup in l_p for p‹∞ uniformly in the mesh, assuming in the two-dimensional case that the triangulations are of Delaunay type, and with a logarithmic bound for p=∞. By a different argument based on a weighted norm estimate for a discrete Green's function this is improved to hold without a logarithmic factor for p=∞ in one dimension under a weak mesh-ratio condition. Our estimates are applied to show stability also for time stepping methods.

In the present paper monotone difference schemes of the second order of approximation and accuracy for differential boundary-value problems of the second and third kind without using the basic differential equation at the domain of the boundary are constructed. The main idea is based on the assumption of the existence and uniqueness of a smooth solution in some sufficiently small neighborhood of the definition domain of the problem and the use of only half-integer nodes of the grid (boundary points are excluded from the calculated nodes). In this case, the boundary conditions are directly approximated with the second order on a two-point stencil. If we assume that the equation has a meaning at the boundary nodes as well, then in this case monotone schemes of the fourth order of accuracy have been constructed. It is shown that in the case of Neumann problem it is necessary to construct such computational procedures, which are monotone and satisfy the grid maximum principle with respect to the flow (of the first derivatives with respect to space variables).

A new second-order accurate scheme for the computation of unsteady viscous incompressible flows is proposed. The scheme is based on the vorticity-stream function formulation along the characteristics and consists of combining the modified method of characteristics with an explicit scheme with an extended real stability interval. A comparison of the new method with the semi-Lagrangian Cranck-Nicolson and classical semi-Lagrangian Runge-Kutta schemes is presented. Numerical results are carried out on Navier-Stokes equations and this efficient second-order scheme has also made it possible to compute the driven cavity ow at a high Reynolds number on a refined grid at a reasonable cost. The procedure can be generalized to more than two dimensions.

To obtain an approximate solution of the steady-state convectiondiffusion problem, it is necessary to solve the corresponding system of linear algebraic equations. The basic peculiarity of these LA systems is connected with the fact that they have non-symmetric matrices. We discuss the questions of approximate solution of 2D convection-diffusion problems on the basis of two- and three-level iterative methods. The general theory of iterative methods of solving grid equations is used to present the material of the paper. The basic problems of constructing grid approximations for steady-state convection-diffusion problems are considered. We start with the consideration of the Dirichlet problem for the differential equation with a convective term in the divergent, nondivergent, and skew-symmetric forms. Next, the corresponding grid problems are constructed. And, finally, iterative methods are used to solve approximately the above grid problems. Primary consideration is given to the study of the dependence of the number of iteration on the Peclet number, which is the ratio of the convective transport to the diffusive one.