  Standard explicit schemes for parabolic equations are not very convenient for
computing practice due to the fact that they have strong restrictions on a time
step. More promising explicit schemes are associated with explicit-implicit
splitting of the problem operator (Saul'yev asymmetric schemes, explicit
alternating direction (ADE) schemes, group explicit method). These schemes
belong to the class of unconditionally stable schemes, but they demonstrate bad
approximation properties. These explicit schemes are treated as schemes of the
alternating triangle method and can be considered as factorized schemes where
the problem operator is splitted into the sum of two operators that are adjoint
to each other. Here we propose a multilevel modification of the alternating
triangle method, which demonstrates better properties in terms of accuracy. We
also consider explicit schemes of the alternating triangle method for the
numerical solution of boundary value problems for hyperbolic equations of
second order. The study is based on the general theory of stability
(well-posedness) for operator-difference schemes.
