Band 8, Heft 2

This paper considers two-dimensional hierarchical locally-refined rect-angular meshes with dynamic adaptation to the solution. A parallel algorithm with dynamic load balancing has been developed. Adaptive meshes have been used for the problem of passive contaminant transport in an oil-bearing stratum at water flooding. A model neglecting the capillary and gravity forces (the Buckley — Leverett model) has been used.

The goal of the research is to construct practicable numerical algorithms for stiff systems of ordinary differential equations which let you increase the accuracy of the approximate solution without decreasing the length of the time interval. To achieve this goal, we have constructed a family of new iterative analytic processes generalising the Picard process. For a basic representative of this family, we demon-strate its better convergence properties on a scalar linear problem in comparison with the classical Picard process. For the general form of such iterative processes, we discuss their connection with existing methods for operator equations and propose a method for choosing their parameters. The efficiency of this parameter determination method is justified with a numerical experiment. In conclusion we propose a general approach to the construction of numerical algorithms which is based on the discretisation of the constructed iterative analytic processes.

We first derive the general error estimate for one – dimensional balance laws and afterwards we apply it to the elastodynamics system with source and to the isentropic Euler system with damping.

In this paper, Jacobi and trigonometric polynomials are used to con-struct the approximate solution of a singular integral equation with multiplicative Cauchy kernel in the half-plane.

For the difference scheme approximating the gas dynamics problem in Riemann invariants a priory estimates with respect to the initial data have been obtained. These estimates are proved without any assumptions about the solution of the differential problem using only limitations for the initial and boundary conditions. Estimates of stability in the general case have been obtained only for the finite instant of time. The uniqueness and convergence of the difference solution are also considered. The results of the numerical experiment confirming theoretical results are given.

With the example of solving some known modeling problems the fea-tures of constructing grids adapted to the solution of parabolic equations are consid-ered. Convection-diffusion problems are described by nonlinear Burgers and Buckley — Leverette equations. A detailed analysis of the differential approximations and numerical results shows that the idea of using an arbitrary time-dependent system of coordinates for adapted grid generation in combination with the principle of quasi-stationarity makes the dy-namic adaptation method universal, effective, and algorithmically simple. Universality is achieved due to the use of an arbitrary time-dependent system of co-ordinates that moves at a velocity determined by the unknown solution. This universal approach makes it possible to generate adapted grids for time-dependent problems of mathematical physics with various mathematical features. Among these features are large gradients, and propagation of weak and strong discontinuities in known nonlinear transfer problems. The efficiency is determined by automatically fitting the velocity of the moving nodes to the dynamics of the solution. The close relationship between the adaptation mechanism and the structure of parabolic equations allows one to automatically control nodes motion of nodes so that their trajectories do not intersect. The simplicity of the algorithm is achieved due to the general approach to the adaptive grid generation that is independent of the form and type of the differential equations.

In this paper we estimate and analyze the errors associated with the use of the discrete (fast) Fourier transformation for the numerical calculation of convo-lutions. We suggest and compare methods to reduce these errors without loosing the computational efficiency of the calculation scheme. A typical field of application of our findings is the calculation of aggregate loss distributions for, e.g., losses from insurance cases or operational risk losses in the finance industry.