Современные проблемы математического моделирования: сборник трудов XVIII Всероссийской конференции-школы молодых исследователей (пос. Абрау-Дюрсо, 16–20 сентября 2019 г.)
The boundary value problem for the Poisson and Helmholtz equations with a piecewise constant coefficient with a jump on a triangle is studied numerically. At the jump of the coefficient (at the boundary of the media), the docking conditions are set. A compact difference scheme with high accuracy with a relatively small number of calculations is proposed.
When solving boundary value problems of mathematical physics, compact schemes allow increasing (in comparison with classical ones) the solution accuracy order with a slight increase in the number of arithmetic operations. An indispensable condition of the algorithm is the use of the double-sweep approach. A method for calculating the coefficients of schemes is shown both on stencils combined for the solution and the right side of an equation, and for chess grids (stencils are shifted by half a step). Fourier analysis confirms a high order of approximation.
Algorithms for three-dimensional visualization of atmospheric front surfaces have been developed. Previously, the technology of visualization of intersections of atmospheric fronts with isobaric levels was implemented. These are curves that show the intensity and type of the atmospheric front in color. At this stage, it was necessary a) to find out which curves at different levels correspond to the same two-dimensional surface in three-dimensional space; b) to build approximations of these surfaces. A significant problem for constructing the algorithm: the curves are not submitted analytically, but by pixels on a rectangle. Curves can branch. The developed algorithms use spline-approximation of curves and approximation by Bezier surfaces. The resulting surfaces can be "viewed" on the screen at different spatial scales and at different angles, i.e. simulated "exploration" of the surfaces of atmospheric fronts. The initial information for this technology is the prognostic output of the COSMO-ru scheme. Therefore, the user will be able to monitor the dynamics of the surfaces of atmospheric fronts and their intensity to make operational decisions to optimize the flight trajectory.