Method of alternating directions for a nonlinear diffusion equation with delay

A nonlinear diffusion equation with two spatial variables and several time delay variables is considered. The problem is discretized. Constructions of the alternating directions method with piecewise linear interpolation and extrapolation by continuation are presented. This method has the second order of smallness with respect to the time discretization step $\Delta$ and the space discretization step $h$. As a result, the method is reduced to solving two tridiagonal systems of linear algebraic equations at each time step, which have diagonal dominance. These systems are efficiently solved using the sweep method. The order of the residual without interpolation of the method is studied. Under certain assumptions, the convergence of the method with the order $O(\Delta^2+h^2)$ is justified. The results of numerical modeling for a diffusion equation with two delay variables are presented. The computable orders of convergence for each discretization step in the examples turned out to be close to the theoretically obtained orders of convergence for the corresponding discretization steps.