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Volume 37, Issue 5
A Singular Parameterized Finite Volume Method for the Advection-Diffusion Equation in Irregular Geometries

Chang Yang & Meng Wu

DOI: 10.4208/jcm.1807-m2017-0029

J. Comp. Math., 37 (2019), pp. 579-608.

Published online: 2019-03

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  • Abstract

Solving the advection-diffusion equation in irregular geometries is of great importance for realistic simulations. To this end, we adopt multi-patch parameterizations to describe irregular geometries. Different from the classical multi-patch parameterization method, $C^1$-continuity is introduced in order to avoid designing interface conditions between adjacent patches. However, singularities of parameterizations can't always be avoided. Thus, in this paper, a finite volume method is proposed based on smooth multi-patch singular parameterizations. It is called a singular parameterized finite volume method. Firstly, we present a numerical scheme for pure advection equation and pure diffusion equation respectively. Secondly, numerical stability results in $L^2$ norm show that the numerical method is not suffered from the singularities. Thirdly, the numerical method has second order accurate in $L^2$ norm. Finally, three numerical tests in different irregular geometries are presented to show efficiency of this numerical method.

  • Keywords

Finite volume method, Smooth multi-patch singular parameterizations, The advection-diffusion equation, Irregular geometries.

  • AMS Subject Headings

65D17, 65M08, 65M50

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

yangchang@hit.edu.cn (Chang Yang)

meng.wu@hfut.edu.cn (Meng Wu)

  • BibTex
  • RIS
  • TXT
@Article{JCM-37-579, author = {Yang , Chang and Wu , Meng}, title = {A Singular Parameterized Finite Volume Method for the Advection-Diffusion Equation in Irregular Geometries}, journal = {Journal of Computational Mathematics}, year = {2019}, volume = {37}, number = {5}, pages = {579--608}, abstract = {

Solving the advection-diffusion equation in irregular geometries is of great importance for realistic simulations. To this end, we adopt multi-patch parameterizations to describe irregular geometries. Different from the classical multi-patch parameterization method, $C^1$-continuity is introduced in order to avoid designing interface conditions between adjacent patches. However, singularities of parameterizations can't always be avoided. Thus, in this paper, a finite volume method is proposed based on smooth multi-patch singular parameterizations. It is called a singular parameterized finite volume method. Firstly, we present a numerical scheme for pure advection equation and pure diffusion equation respectively. Secondly, numerical stability results in $L^2$ norm show that the numerical method is not suffered from the singularities. Thirdly, the numerical method has second order accurate in $L^2$ norm. Finally, three numerical tests in different irregular geometries are presented to show efficiency of this numerical method.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1807-m2017-0029}, url = {http://global-sci.org/intro/article_detail/jcm/13036.html} }
TY - JOUR T1 - A Singular Parameterized Finite Volume Method for the Advection-Diffusion Equation in Irregular Geometries AU - Yang , Chang AU - Wu , Meng JO - Journal of Computational Mathematics VL - 5 SP - 579 EP - 608 PY - 2019 DA - 2019/03 SN - 37 DO - http://doi.org/10.4208/jcm.1807-m2017-0029 UR - https://global-sci.org/intro/article_detail/jcm/13036.html KW - Finite volume method, Smooth multi-patch singular parameterizations, The advection-diffusion equation, Irregular geometries. AB -

Solving the advection-diffusion equation in irregular geometries is of great importance for realistic simulations. To this end, we adopt multi-patch parameterizations to describe irregular geometries. Different from the classical multi-patch parameterization method, $C^1$-continuity is introduced in order to avoid designing interface conditions between adjacent patches. However, singularities of parameterizations can't always be avoided. Thus, in this paper, a finite volume method is proposed based on smooth multi-patch singular parameterizations. It is called a singular parameterized finite volume method. Firstly, we present a numerical scheme for pure advection equation and pure diffusion equation respectively. Secondly, numerical stability results in $L^2$ norm show that the numerical method is not suffered from the singularities. Thirdly, the numerical method has second order accurate in $L^2$ norm. Finally, three numerical tests in different irregular geometries are presented to show efficiency of this numerical method.

Yang , Chang and Wu , Meng. (2019). A Singular Parameterized Finite Volume Method for the Advection-Diffusion Equation in Irregular Geometries. Journal of Computational Mathematics. 37 (5). 579-608. doi:10.4208/jcm.1807-m2017-0029
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