任意四边形网格上扩散问题的一个稳定九点格式

doi:10.12288/szjs.2019.1.51

数值计算与计算机应用 ›› 2019, Vol. 40 ›› Issue (1): 51-67.DOI: 10.12288/szjs.2019.1.51

任意四边形网格上扩散问题的一个稳定九点格式

洪旗, 苏帅

中国工程物理研究院研究生院, 北京 100088

收稿日期:2018-10-29 出版日期:2019-03-15 发布日期:2019-03-15
基金资助:
国家自然科学基金（11871009，11771052）.

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洪旗, 苏帅. 任意四边形网格上扩散问题的一个稳定九点格式[J]. 数值计算与计算机应用, 2019, 40(1): 51-67.

Hong Qi, Su Shuai. A STABLE NINE-POINT SCHEME FOR DIFFUSION PROBLEMS ON ARBITRARY QUADRILATERAL MESHES[J]. Journal of Numerical Methods and Computer Applications, 2019, 40(1): 51-67.

A STABLE NINE-POINT SCHEME FOR DIFFUSION PROBLEMS ON ARBITRARY QUADRILATERAL MESHES

Hong Qi, Su Shuai

Graduate School of China Academy of Engineering Physics, Beijing 100088, China

Received:2018-10-29 Online:2019-03-15 Published:2019-03-15

通过修正Q₁有限体积元方法处理节点未知量，本文提出且分析了扩散问题的一个稳定的九点格式.基于修正Q₁有限体积元格式理论和离散泛函分析，我们在弱几何条件给出了稳定性分析和H¹误差估计.与已有的一些中心型和杂交型格式相比，该格式不遭受所谓的数值热障现象.但是该格式需要多求解一次线性方程组.数值实验表明了格式有效性并且验证了理论分析.

关键词： 扩散问题, 九点格式, 修正Q₁有限体积元, 稳定性和H¹误差估计

We propose and analyze a stable nine-point scheme for diffusion problems by applying a modified Q1 finite volume element method (mQ1-FVEM) to treat the vertex unknowns. Based on the theoretical results of the mQ₁-FVEM and a simple discrete functional technique, the stability result and error estimate of the resulting nine-point scheme both in H¹ norm are obtained under a standard and weak geometric assumption. Our proposed scheme does not possess the so-called numerical heat-barrier issue suffered by many existing cell-centered or hybrid schemes. But, we need to solve two linear algebraic systems. Numerical experiments are also presented to show the efficiency of the proposed scheme and validate the theoretical analysis.

Key words: diffusion problem, nine-point scheme, Q₁ finite volume element, stability and H¹ error estimate

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任意四边形网格上扩散问题的一个稳定九点格式

A STABLE NINE-POINT SCHEME FOR DIFFUSION PROBLEMS ON ARBITRARY QUADRILATERAL MESHES

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