The use of pseudoresiduals in the study of convergence of unstable difference boundary value problems for linear nonhomogeneous ordinary second-order differential equations

The paper considers the previously proposed method of numerical integration using the matrix calculus in the study of boundary value problems for nonhomogeneous linear ordinary differential equations of the second order with variable coefficients. According to the indicated method, when compiling a system of difference equations, an arbitrary degree of the Taylor polynomial in expanding the unknown solution of the problem into a Taylor series can be chosen while neglecting the approximation of the derivatives by finite differences.
Some aspects of the convergence of an unstable second-order difference boundary value problem are investigated. The concept of a pseudo-residual on a certain vector is introduced for an ordinary differential equation. On the basis of the exact solution of the difference boundary value problem, an approximate solution has been built, where the norm of pseudo-residuals is different from the trivial value.
It has been established theoretically that the estimate of the pseudo-residual norm decreases with an increase in the used degree of the Taylor polynomial and with a decrease in the mesh discretization step. The definitions of conditional stability and conditional convergence are given; a theoretical connection between them is established. The perturbed solution has been built on the basis of the found vector of pseudo-residuals, the estimate of the norm of its deviation from the exact solution of the difference boundary value problem has been calculated, which allows one to identify the presence of conditional stability. A theoretical relationship between convergence and conditional convergence is established.
The results of numerical experiments are presented.