Hexagonal grid approximation of the solution of the heat equation on special polygons

Abstract

We consider the first type boundary value problem of the heat equation in two space dimensions on special polygons with interior angles alpha(j)pi, j = 1,2, ..., M, where alpha(j) is an element of {1/2, 1/3, 2/3}. 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 both difference problems are unconditionally stable. It is also shown that the solutions of the constructed Difference Problem 1 and Difference Problem 2 converge to the exact solution on the grids of order O(h(2) + tau(2)) and O(h(4) + tau) respectively, where h and root 3/2 are the step sizes in space variables x(1) and x(2) respectively and tau is the step size in time. Furthermore, theoretical results are justified by numerical examples on a rectangle, trapezoid and parallelogram.