A novel monotonous finite volume element scheme for convection dominant diffusion problem
Abstract
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.
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BibTeX
@misc{nie2026novel,
title = {{A novel monotonous finite volume element scheme for convection dominant diffusion problem}},
author = {Cunyun Nie and Xiaoling Chen and Zhujun wang and Zhikun Tian and Chengjie Xia},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23546},
eprint = {2609.23546},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23546},
}
RIS
TY - UNPB
TI - A novel monotonous finite volume element scheme for convection dominant diffusion problem
AU - Cunyun Nie
AU - Xiaoling Chen
AU - Zhujun wang
AU - Zhikun Tian
AU - Chengjie Xia
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23546
UR - https://arxiv.org/abs/2609.23546
AB - 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.
ER -