Volume 10, Issue 1
A Maximum-Principle-Preserving Third Order Finite Volume SWENO Scheme on Unstructured Triangular Meshes

Yunrui Guo, Lingyan Tang, Hong Zhang & Songhe Song

Adv. Appl. Math. Mech., 10 (2018), pp. 114-137.

Published online: 2018-10

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  • Abstract

We modify the construction of the third order finite volume WENO scheme on triangular meshes and present a simplified WENO (SWENO) scheme. The novelty of the SWENO scheme is the less complexity and lower computational cost when deciding the smoothest stencil through a simple mechanism. The LU decomposition with iterative refinement is adopted to implement ill-conditioned interpolation matrices and improves the stability of the SWENO scheme efficiently. Besides, a scaling technique is used to circument the growth of condition numbers as mesh refined. However, weak oscillations still appear when the SWENO scheme deals with complex low density equations. In order to guarantee the maximum-principle-preserving (MPP) property, we apply a scaling limiter to the reconstruction polynomial without the loss of accuracy. A novel procedure is designed to prove this property theoretically. Finally, numerical examples for one- and two-dimensional problems are presented to verify the good performance, maximum principle preserving, essentially non-oscillation and high resolution of the proposed scheme.

  • Keywords

Triangular meshes, WENO, scaling limiter, maximum-principle-preserving.

  • AMS Subject Headings

65M10, 78A48

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-10-114, author = {}, title = {A Maximum-Principle-Preserving Third Order Finite Volume SWENO Scheme on Unstructured Triangular Meshes}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {10}, number = {1}, pages = {114--137}, abstract = {

We modify the construction of the third order finite volume WENO scheme on triangular meshes and present a simplified WENO (SWENO) scheme. The novelty of the SWENO scheme is the less complexity and lower computational cost when deciding the smoothest stencil through a simple mechanism. The LU decomposition with iterative refinement is adopted to implement ill-conditioned interpolation matrices and improves the stability of the SWENO scheme efficiently. Besides, a scaling technique is used to circument the growth of condition numbers as mesh refined. However, weak oscillations still appear when the SWENO scheme deals with complex low density equations. In order to guarantee the maximum-principle-preserving (MPP) property, we apply a scaling limiter to the reconstruction polynomial without the loss of accuracy. A novel procedure is designed to prove this property theoretically. Finally, numerical examples for one- and two-dimensional problems are presented to verify the good performance, maximum principle preserving, essentially non-oscillation and high resolution of the proposed scheme.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2016-0196}, url = {http://global-sci.org/intro/article_detail/aamm/10504.html} }
TY - JOUR T1 - A Maximum-Principle-Preserving Third Order Finite Volume SWENO Scheme on Unstructured Triangular Meshes JO - Advances in Applied Mathematics and Mechanics VL - 1 SP - 114 EP - 137 PY - 2018 DA - 2018/10 SN - 10 DO - http://dor.org/10.4208/aamm.OA-2016-0196 UR - https://global-sci.org/intro/aamm/10504.html KW - Triangular meshes, WENO, scaling limiter, maximum-principle-preserving. AB -

We modify the construction of the third order finite volume WENO scheme on triangular meshes and present a simplified WENO (SWENO) scheme. The novelty of the SWENO scheme is the less complexity and lower computational cost when deciding the smoothest stencil through a simple mechanism. The LU decomposition with iterative refinement is adopted to implement ill-conditioned interpolation matrices and improves the stability of the SWENO scheme efficiently. Besides, a scaling technique is used to circument the growth of condition numbers as mesh refined. However, weak oscillations still appear when the SWENO scheme deals with complex low density equations. In order to guarantee the maximum-principle-preserving (MPP) property, we apply a scaling limiter to the reconstruction polynomial without the loss of accuracy. A novel procedure is designed to prove this property theoretically. Finally, numerical examples for one- and two-dimensional problems are presented to verify the good performance, maximum principle preserving, essentially non-oscillation and high resolution of the proposed scheme.

Yunrui Guo, Lingyan Tang, Hong Zhang & Songhe Song. (2020). A Maximum-Principle-Preserving Third Order Finite Volume SWENO Scheme on Unstructured Triangular Meshes. Advances in Applied Mathematics and Mechanics. 10 (1). 114-137. doi:10.4208/aamm.OA-2016-0196
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