On the solution of elliptic equations by the ray variable method
Abstract:
A method for solving a boundary value problem for a linear elliptic equation of the second order is described using reduction to a boundary value problem for a system of kinetic equations. Kinetic equations have characteristics, i.e. rays crossing a region. Due to the transition to ray variables, the solution to the original multidimensional problem is replaced by the solution of a series of one-dimensional problems at the rays. A finite-analytic scheme for numerically solving problems in a domain with discontinuous coefficients and sources is constructed. The region is divided into cells, within which the coefficients and sources are continuous, and the finite discontinuities (if any) occur at the cell boundaries. Next, the solutions are cross-linking at the cell boundaries. In the scheme, there is no strong dependence of the accuracy of approximation from the size and shape of the cells that inherent in finite difference schemes. To solve a small-sized system of the scheme algebraic equations, direct (non-iterative) algorithms of computational algebra can be used, such as the Gaussian elimination technique, and the tridiagonal matrix algorithm.
Keywords:
elliptic equations, boundary value problems, method of ray variables, finite-analytic schemes
Publication language:russian, pages:36
Research direction:
Mathematical modelling in actual problems of science and technics
