一种新的对流占优扩散问题的单调有限体积元格式
A novel monotonous finite volume element scheme for convection dominant diffusion problem
- Hunan Institute of Engineering（湖南工程学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出一种新的单调有限体积元格式，通过引入迎风体积和渐近展开构造二阶离散算子，用于求解对流占优扩散问题，并验证了误差的L2范数收敛性。
AI中文摘要：
针对对流占优扩散问题，提出了一种新的单调有限体积元（MFVE）格式。本文的主要贡献包括四个方面。首先，类似于迎风单元，引入一个迎风子控制体，称为迎风体积，用于对流项的离散，从而获得迎风性质。其次，通过在迎风体积中进行数值积分，直接得到平衡方程中对流项的一阶线性离散算子，这与经典有限体积元方法不同。第三，通过一个扩展插值函数，推导出每个节点对偶单元中梯度函数的渐近展开式。然后，利用该渐近展开式及其有限体积元解的误差估计，构造对流项的二阶非线性离散算子，其中一些扰动系数被巧妙地放入展开式中，作为高阶修正。第四，设计了一种新的二阶MFVE格式，同时伴随另一种一阶MFVE格式，用于对流占优扩散问题，并配以保正的有限体积元（PFVE）离散扩散算子。推导了近似解误差的L2范数。最后，数值结果验证了理论结论。
英文摘要：
One novel monotonous finite volume element (MFVE) scheme is put forward for convection dominant diffusion problem. The main contributions of this paper include four aspects. Firstly, one upwind sub-control volume, called upwind volume, is introduced for the discretization of the convection term similar to the upwind element, which leads to the upwind property. Secondly, one first-order linear discrete operator for the convection term in the balance equation is directly obtained by numerical integration in upwind volume, which is different from the classical finite volume methods.Thirdly, one asymptotic expansion of gradient functions in the dual element of each node is derived by one extension interpolation function. Then, one second-order nonlinear discrete operator for the convection term is constructed by the asymptotic expansion and error estimation of its FVE solution, where some perturbed coefficients are skillfully put into the expansion, which serves as a high-order correction. Fourthly, one novel two-order MFVE scheme, also accompanied by another one-order MFVE, is designed for convection-dominated diffusion problems together with one positivity-preserving finite volume element (PFVE) discrete diffusive operator. The L2 norm of the error of the approximate solution is derived. Finally, numerical results validate theoretical conclusions.
