About the iteration method for solving difference equations

The new iteration line-by-line method with a variable compensation parameter is proposed for solving a system of difference equations that arises from the implicit approximation of two-dimensional elliptic and parabolic differential equations. Calculations have shown that the method's convergence rate is insensitive to the variation of coefficients at high derivatives in differential equation and weakly depends on a quantity of grid nodes. The use of computational technology is especially effective on detailed grids and allows to reduce a necessary number of iteration steps up to two orders of magnitude as compared with the initial line-by-line method.