Homogeneous schemes for second-order equations with variable coefficients

1 HOMOGENEOUS SCHEMES FOR SECOND-ORDER EQUATIONS WITH VARIABLE COEFFICIENTS 

Introduction. Modern computers permit implementation of highly accurate difference schemes. Just for this reason, it is unreasonable to develop difference methods and create high quality software for solving particular problems. An actual problem consists of constructing difference schemes capable of describing classes of problems that are determined by a given type 

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For instance, of great interest are through or continuous execution schemes available for solving the diffusion equation with discontinuous diffusion coefficients by means of the same formulae (software). No selection of points or lines of discontinuities of the coefficients applies here. This means that the scheme remains unchanged in a neighborhood of discontinuities and the computations at all grid nodes can be carried out by the same formulae without concern of discontinuity or continuity of the diffusion coefficient. 

Homogeneous through execution schemes are quite applicable in the cases where the diffusion coefficient is found as an approximate solution of other equations. For instance, such schemes are aimed at solving the equations of gas dynamics in a heat conducting gas when the diffusion coefficient depends on the density and has discontinuities on the shock waves. 

In the theory of difference schemes with a primary family of schemes the coefficients of a homogeneous difference scheme are expressed through the coefficients of the initial differential equation by means of the so-called pattern functionals the arbitrariness in the choice of these functionals is limited by the requirements of approximation, solvability, etc. There are various ways of taking care of these restrictions. The availability of a primary family of homogeneous difference schemes is ensured by a family of admissible pattern functionals known in advance. 

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