Construction and study of high-order accurate schemes for solving the one-dimensional heat equation

The method of undetermined coefficients on multipoint stencils with two time levels was used to construct compact difference schemes of $O(\tau^3,h^6)$ accuracy intended for solving boundary value problems for the one-dimensional heat equation. The schemes were examined for von Neumann stability, and numerical experiments were conducted on a sequence of grids with mesh sizes tending to zero. One of the schemes was proved to be absolutely stable. It was shown that, for smooth solutions, the high order of convergence of the numerical solution agrees with the order of accuracy; moreover, solutions accurate up to $\sim10^{-12}$ are obtained on grids with spatial mesh sizes of $\sim10^{-2}$. The formulas for the schemes are rather simple and easy to implement on a computer.