Analytical and numerical investigation of the spectra of three-point difference operators

The properties of the spectra of discrete spatially one-dimensional problems of convection — diffusion type with constant coefficients and nonstandard boundary conditions are examined in the framework of stability of explicit algorithms for time-dependent problems of mathematical physics. An analytical method is proposed for finding isolated limit points of the operator spectrum. Limit points are determined for the difference transport equation with different versions of nonreflecting boundary conditions and for an approximation of the heat conduction equation on a grid with condensation near the boundary. Stability and other properties of the spectrum are also established numerically. __________ Translated from Prikladnaya Matematika i Informatika, No. 27, pp. 25–45, 2007.