In this article, the one-dimensional parabolic equation with three types of integral nonlocal boundary conditions is approximated by the implicit Euler finite difference scheme. Stability analysis is done in the maximum norm and it is proved that the radius of the stability region and the stiffness of the discrete scheme depends on the signs of coefficients in the nonlocal boundary condition. The known stability results are improved. In the case of a plain integral boundary condition, the conditional convergence rate is proved and the regularization relation between discrete time and space steps is proposed. The accuracy of the obtained estimates is illustrated by results of numerical experiments.

: This paper presents the analysis of the split step solvers for multidimensional Schrödinger problems. The second-order symmetrical splitting techniques are applied. The standard operator splitting is used to split the linear diffraction and reaction/potential processes. The dimension splitting exploits the commuting property of one-dimensional discrete diffraction operators. Alternating Direction Implicit (ADI) and Locally One-Dimensional (LOD) algorithms are constructed and stability is investigated for two- and three-dimensional problems. Compact high-order approximations are applied to discretize diffraction operators. Results of numerical experiments are presented and convergence of finite difference schemes is investigated.