Hexagonal grid approximation of the solution of the heat equation on special polygons
Abstract
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 $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>α</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mi>π</mml:mi></mml:math> , $j=1,2,\ldots,M$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>M</mml:mi></mml:math> , where $\alpha _{j}\in \{ \frac{1}{2},\frac{1}{3},\frac{2}{3} \} $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>α</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>∈</mml:mo><mml:mo>{</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac><mml:mo>,</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac><mml:mo>,</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac><mml:mo>}</mml:mo></mml:math> . 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} ) $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:msup><mml:mi>h</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mi>τ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>)</mml:mo></mml:math> and $O ( h^{4}+\tau ) $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:msup><mml:mi>h</mml:mi><mml:mn>4</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mi>τ</mml:mi><mml:mo>)</mml:mo></mml:math> respectively, where h and $\frac{\sqrt{3}}{2}h $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mfrac><mml:msqrt><mml:mn>3</mml:mn></mml:msqrt><mml:mn>2</mml:mn></mml:mfrac><mml:mi>h</mml:mi></mml:math> are the step sizes in space variables $x_{1}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>x</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math> and $x_{2}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math> respectively and τ is the step size in time. Furthermore, theoretical results are justified by numerical examples on a rectangle, trapezoid and parallelogram.