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 α j π, j = 1, 2,. .. , M, where α j ∈ { 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 + τ 2) and O(h 4 + τ) respectively, where h and √ 3 2 h are the step sizes in space variables x 1 and x 2 respectively and τ is the step size in time. Furthermore, theoretical results are justified by numerical examples on a rectangle, trapezoid and parallelogram.

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

“…For the numerical solution of the BVP(1) the following difference problem (named as Difference Problem 1) was given in [29] which we will consider as the Stage 1 H 2nd of the two stage implicit method:…”

“…By numbering the interior grid points using standard ordering as L j , j = 1, 2, ..., N, the obtained algebraic linear system of equations in matrix form given in [29] is as follows:…”

“…From (10)- (13) and (26), the error function ε u h,τ satisfies the following system as given in [29]…”

“…The exhibited linear and spiral waves were more efficient than similar computation carried out on rectangular finite volume schemes. Furthermore, hexagonal grids were applied to approximate the solution of the first type boundary value problem of the heat equation in [28][29][30], convection-diffusion equation in [31], and Dirichlet type boundary value problem of the two dimensional Laplace equation in [32]. In the recent study [29], the solution of first type boundary value problem of heat equation…”

“…, and D is the closure of D. In addition, hexagonal grid structure and basic notations are given. Further, at the first stage, a two layer implicit method on hexagonal grids given in [29] with O h 2 + τ 2 order of accuracy, where h and √ 3…”