An iterative alternating-directions method for the poisson difference equation in curvilinear orthogonal coordinates

On high-order compact difference schemes

“…Analogously we can consider approximations on the other Solutions, which are expanded in an even power series of r and are characteristic of problems of mathematical physics (see [3]). To conclude this section we emphasize that high-order schemes analogous to (3.3) were considered in [3,17].…”

“…To conclude this section we emphasize that high-order schemes analogous to (3.3) were considered in [3,17]. However, there was a difference in the determination of the coordinate lines r = r\, and the peculiarities of approximation along the axis were not taken into account (the domain with r > R > 0 is considered in [17] and the Dirichlet condition for r = 0 in [3], which significantly affect the formulation of the problem). Here n is the number of a time step, I is the unit operator, v% = {υ·}}, τ η = t n -* n _i is a time step.…”

“…In this case the used procedure of expansions in terms of harmonic fimctions results in such cumbersome formulae for coefficients that it becomes essentially impossible to study the properties of difference equations. In [3,17] nine-point approximations for the same equations were constructed in the case r > r 0 > 0. We should also mention the works [20,21,23], where stationary and nonstationary boundary value problems of a diffusion type with variable coefficients were studied.…”