Two Stage Implicit Method on Hexagonal Grids for Approximating the First Derivatives of the Solution to the Heat Equation

The first type of boundary value problem for the heat equation on a rectangle is considered. We propose a two stage implicit method for the approximation of the first order derivatives of the solution with respect to the spatial variables. To approximate the solution at the first stage, the unconditionally stable two layer implicit method on hexagonal grids given by Buranay and Arshad in 2020 is used which converges with $O\left(h^2+{\tau }^2\right)$ of accuracy on the grids. Here, h and $\frac{\sqrt{3}}{2}h$ are the step sizes in space variables $x_1$ and $x_2$, respectively and $\tau$ is the step size in time. At the second stage, we propose special difference boundary value problems on hexagonal grids for the approximation of first derivatives with respect to spatial variables of which the boundary conditions are defined by using the obtained solution from the first stage. It is proved that the given schemes in the difference problems are unconditionally stable. Further, for $r=\frac{\omega \tau }{h^2}\le \frac{3}{7}$, uniform convergence of the solution of the constructed special difference boundary value problems to the corresponding exact derivatives on hexagonal grids with order $O\left(h^2+{\tau }^2\right)$ is shown. Finally, the method is applied on a test problem and the numerical results are presented through tables and figures.