Homogeneous difference schemes

This chapter presents the theory of homogeneous difference schemes for the solution of equations with variable coefficients... 

A special attention is being paid to various forms of homogeneous difference schemes and their approximation and convergence for discontinuous coefficients k, q and / as well as for non-equidistant grids. Not mnch is known in such cases. Later in this chapter we will survey some devices that can be used to obtain simpler forms and higher orders. 

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. 

The principal question in the theory of homogeneous difference schemes is connected with further design of admissible schemes within a primary family for solving a class of typical problems as wide as possible and choosing the most efficient ones (in accuracy, volume of computations, etc.). 

The integro-interpolational method for constructing homogeneous difference schemes. Various physical processes (heat conduction or diffusion, vibrations, gas dynamics, etc.) are well-characterized by some integral... 

The third boundary-value problem. The main goal of our studies is a homogeneous difference scheme for the boundary-value problem of the third kind ... 

---