Classification of Difference Schemes of Maximum Possible Accuracy on Extended Symmetric Stencils for the Schrodinger Equation and the Heat Conduction Equation

All possible symmetric two-level difference schemes on arbitrary extended stencils are considered for the Schrodinger equation and for the heat conduction equation. The coefficients of the schemes are found from conditions under which the maximum possible order of approximation with respect to the main variable is attained. A class of absolutely stable schemes is considered in a set of maximally exact schemes. To investigate the stability of the schemes, the von Neumann criterion is verified numerically and analytically. It is proved that the schemes are absolutely stable or unstable depending on the order of approximation with respect to the evolution variable. As a result of the classification, absolutely stable schemes up to the tenth order of accuracy with respect to the main variable have been constructed.