ABSTRACT
This chapter considers new notions and ideas in regard to the theory of finite-difference schemes, which are related to the heat equation. Methods with a variable time step are usually used when the coefficients of the differential equations have some singularity as the functions of time. For finite-difference equations with variable coefficients, the method of harmonics is used in combination with the method of frozen coefficients. In the differential case, the heat equation satisfies the so-called maximum principle. The statement of the maximum principle for the initial-boundary value problem is: the maximum and minimum of the solution for the heat equation can be reached either at the initial moment or at the boundary of the domain. For Robin boundary conditions we will consider only implicit finite-difference schemes. The approximations for these conditions in the case of the heat equation look similar to the case for the elliptic equation.
