Homogeneous Difference Schemes
This chapter presents the theory of homogeneous difference schemes for the
solution of equations with variable coefficientsLu+f(x)=O, Lu= dx d ( k(x) du) dx -g(x)u.
A special attention is being paid to various forms of homogeneous differ-
ence schemes and their approximation and convergence for discontinuous
coefficients k, q and f as well as for non-equidistant grids. Not much is
known in such cases. Later in this chapter we will survey some devices that
can be used to obtain simpler forrns and higher orders.3.1 HOMOGENEOUS SCHEMES FOR SECOND-ORDER EQUATIONS
WITH VARIABLE COEFFICIENTS- Introduction. Modern computers permit implementation of highly accu-
rate 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 ca-
pable of describing classes of problems that are determined by a given type
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