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

All possible symmetric two-level difference schemes on arbitraryextended stencils are considered for the Schrödinger equation and forthe heat conduction equation. The coefficients of the schemes are foundfrom conditions under which the maximum possible order of approximationwith respect to the main variable is attained. A class of absolutelystable schemes is considered in a set of maximally exact schemes. Toinvestigate the stability of the schemes, the von Neumann criterion isverified numerically and analytically. It is proved that the schemes areabsolutely stable or unstable depending on the order of approximationwith respect to the evolution variable. As a result of theclassification, absolutely stable schemes up to the tenth order ofaccuracy with respect to the main variable have been constructed.