Classification of difference schemes of the maximum possible accuracy on extended symmetric stencils for the Schrödinger equation and the heat transfer equation

We study all possible symmetric two-level difference schemes on arbitrary extended stencils for the Schrödinger equation and for the heat conductivity equation. We find the coefficients of the schemes from the conditions under which a maximum possible order of approximation on the main variable is attained. From a set of maximally exact schemes, a class of absolutely stable schemes is isolated. To investigate the stability of the schemes, the Neumann criterion is numerically and analytically verified.
It is proved that the property of schemes to be absolutely stable or unstable significantly depends on the order of approximation on the evolution variable. As a result of the classification it was possible to construct absolutely stable schemes up to the tenth order of accuracy on the main variable.