Hexagonal Grid Approximation of the Solution of Two Dimensional Heat Equation

We consider the first type boundary value problem of heat equation ¶u ¶t = w ¶2u ¶x2 1 + ¶2u ¶x2 2 + f (x1;x2; t) in two space dimensions on special polygons with interior angles ajp, j = 1;2; :::;M, where aj 2 12 ; 1 3 ; 2 3 ;w is positive constant and f is the heat source. To approximate the solution we develop two difference problems on hexagonal grids using two layers with 14 points. It is proved that the given implicit schemes in Difference Problem 1 and Difference Problem 2 are unconditionally stable. We also show that the convergence of the given difference problems to the exact solution are the order of O 􀀀���� h2+t2 and O 􀀀���� h4+t respectively on the grids, where h and p 3 2 h are the step sizes in space variables x1 and x2 respectively and t is the step size in time. The theoretical results are justified by numerical examples on rectangle, trapezoid and parallelogram. Furthermore, a two layer implicit method on hexagonal grids is also proposed for approximating the solution to first type boundary value problem of the heat equation ¶u ¶t = w ¶2u ¶x2 1 + ¶2u ¶x2 2 􀀀����bu+ f (x1;x2; t) on rectangle where w > 0; b 0 are constants and f is the heat source. For the hexagonal grids that have centers h2 units away from the sides of the rectangle at time moment t with one of the neighboring point in the pattern emerging through the sides a special scheme is given. The unconditional stability of the implicit scheme and the convergence of the approximate solution having order O 􀀀���� h4+t2 where h and p 3 2 h are the step sizes in space variables x1 and x2 respectively and t being the step size in time, are proved. The method is applied on test problems and the obtained numerical results justify the given theoretical results. Keywords: Finite difference method, Hexagonal grid, Stability analysis, Error bounds, Two dimensional heat equation.