Band 7, Heft 2

A further development of the dynamic adaptation method for gas dy- namic problems describing multiple interactions of the shock waves, rarefaction waves, and contact boundaries are considered. By the test Woodward — Colella problem the efficiency of the proposed method for solving problems of gas dynamics with explicit definition of shock waves and contact boundaries is shown. For the problem solution mechanism of the adaptation of diffusion type is used. A choice of the adaptation coeffi- cient for obtaining solution in each of subdomains of quasiuniform grid is substantiated. The discontinuities interaction between each other is solved by the Riemann problem. Application of the dynamic adaptation method allowed obtaining solution in 420 cells which practically coincides with the results of WENO5m method in 12800 cells.

The performance of oil filters used in the automotive industry can be significantly improved, especially when computer simulation is an essential component of the design process. In this paper, we consider parallel numerical algorithms for solving mathematical models describing the process of filtration, filtering solid particles out of liquid oil. The Navier — Stokes — Brinkmann system of equations is used to describe the laminar flow of incompressible isothermal oil. The space discretization in the complicated filter geometry is based on the finite-volume method. Special care is taken for an accurate approximation of the velocity and pressure on the interface between the fluid and the porous media. The time discretization used here is a proper modification of the fractional time step discretization (cf. Chorin scheme) of the Navier- Stokes equations, where the Brinkmann term is considered in both the prediction and the correction substeps. A data decomposition method is used to develop a parallel algorithm, where the domain is distributed among the processors by using a structured reference grid. The MPI library is used to implement the data communication part of the algorithm. A theoretical model is proposed for the estimation of the complexity of the given parallel algorithm and a scalability analysis is done on the basis of this model. The results of the computational experiments are presented, and the accuracy and efficiency of the parallel algorithm is tested on real industrial geometries.

The third-kind boundary-value problem for a second-order elliptic equation on a polygonal domain with variable coefficients, mixed derivatives, and first-order terms is approximated by a linear finite element method with first-order accurate quadrature. The corresponding bilinear form does not need to be strongly positive. The discretisation is equivalent to a finite difference scheme. Although the discretisation is in general only first-order consistent, supraconvergence, i.e., convergence of higher order, is shown to take place even on nonuniform grids. If neither oblique boundary sections nor mixed derivatives occur, then the optimal order s is achieved. The supraconvergence result is equivalent to the supercloseness of the gradient.

Numerical methods for simultaneous solution of geofiltration and ge- omigration problems in multilayer systems in contaminant transport investigation have been developed. A multilayer system consists of several aquifers separated by weakly permeable layers. Mathematical models are constructed under the following assump- tion: longitudinal flows predominate in the aquifer and crossflows — in the separation layers. The mass transfer is described by the convection and diffusion equations. The implicit finite volume difference schemes were used. The computational al- gorithm is based on the concept of splitting the schemes on physical processes of two- dimensional filtration and contaminant transport along aquifers and vertical transport through the separation levels. An iterative method of the block Gauss — Seidel type for the realization of implicit finite difference schemes for geofiltration and geomigration problems in the presence of wells and contaminant sources has been constructed. The method is based on the perturbation transfer from cells with sources (wells and sources of contaminant). The iterative process for update of unknowns has been constructed taking into account the position of a cell relative to the calculated cells. The results of the numerical experiment on modeling groundwater flows redistri- bution and contaminant migration in system of two aquifers with operating wells are presented.

The initial boundary-value problem for the two-dimensional quasi-linear heat conduction equation is considered. A new difference scheme approximating the equation is constructed. The main difference between the method proposed in the present paper and the other difference schemes is that for the traveling wave solutions the constructed scheme is exact if the grid steps satisfy certain conditions. The iteration method is used to solve a nonlinear difference equation.