Three-layer of compact difference schemes for the parabolic equation

This paper is dedicated to construction and study of three-layer of compact difference schemes for linear and quasi-linear parabolic equations of order $O(h^4+\tau^2)$. In the linear case, a priori stability estimates from the initial data on the right side are obtained. The basic scheme for constructing difference schemes of a given quality is the asymptotic stability of the second order of accuracy $O(h^2+\tau^2)$ by A. A. Samarsky. The results are generalized to the case of boundary conditions of the third kind, variable coefficients. A three-layer scheme of approximation order $O(h^6+\tau^3)$ is also constructed on a three-point stencil in space, which allows to use an economical sweep method to solve the corresponding system of algebraic equations. Numerical experiments are presented to illustrate the correctness of our theoretical conclusions. Simulation of nonlinear processes with traveling waves showed that these algorithms can also be used for differential problems that have features in solution.