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

Advances in Difference Equations

article

Buranay, Suzan C.¹  Arshad, Nouman¹ 

[1] Department of Mathematics, Eastern Mediterranean University

关键词: Finite difference method;    Hexagonal grid;    Stability analysis;    Error bounds;    Two dimensional heat equation;   

DOI  :  10.1186/s13662-020-02749-z

学科分类：航空航天科学

来源: SpringerOpen

【 摘 要 】

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 $, $j=1,2,\ldots,M$, where $\alpha _{j}\in \{ \frac{1}{2},\frac{1}{3},\frac{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 $\frac{\sqrt{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.

【 授权许可】

CC BY