*(English)*Zbl 1196.65154

A multidimensional parabolic equation can be solved numerically very efficiently using alternating direction implicit (ADI) schemes. The authors expand previous results of numerical analysis for these schemes, namely the unconditional convergence of such numerical solutions in the maximum norm of order two in 2D space and time is proved under a smoothness assumption for the continuous solution of the problem. Moreover, using an asymptotic expansion of the difference solution, a fourth order, both in space and time, approximation is obtained by one Richardson extrapolation.

In the second part the authors investigate high-order compact ADI schemes. The proposed technique can be also used for approximation of this type. A maximum norm error of order $O({\tau}^{2}+{h}_{1}^{4}+{h}_{2}^{4})$, where $\tau $ is the time step and ${h}_{1},{h}_{2}$ represent the grid size, for a 2D space domain is proved. Again by one extrapolation, a sixth order accurate approximation under the condition that the time step is proportional to the squares of the spatial size is derived. A finally presented numerical example confirms the theoretical results of the scheme.

##### MSC:

65M15 | Error bounds (IVP of PDE) |

65M06 | Finite difference methods (IVP of PDE) |

65M12 | Stability and convergence of numerical methods (IVP of PDE) |

35K20 | Second order parabolic equations, initial boundary value problems |

65F10 | Iterative methods for linear systems |