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Volume 17, Issue 4
A Finite Volume Simple WENO Scheme for Convection-Diffusion Equations

Pufan Deng, Gang Peng, Xinlong Feng, Dongmi Luo & Yibing Chen

DOI: 10.4208/nmtma.OA-2024-0003

Numer. Math. Theor. Meth. Appl., 17 (2024), pp. 882-903.

Published online: 2024-12

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  • Abstract

A simple weighted essentially non-oscillatory (SWENO) scheme for solving the convection-diffusion equation is proposed in this paper, where a seventh-order SWENO method and the third-order strong stability preserving (SSP) Runge-Kutta method are adopted for discretizing the space and time, respectively. Then the hyperbolic and diffusive part can achieve the seventh- and sixth-order accuracy, respectively. The proposed method has the following advantages. Firstly, negative linear weights are avoided. Secondly, one reconstruction with one stencil is required for the computation of convective and diffusive fluxes. Finally, the new method does not require the transformation while the diffusion coefficients are degenerate. Numerical examples demonstrate that the new method can achieve sixth-order accuracy in the smooth region and guarantee non-oscillatory properties for the discontinuous problems for one- and two-dimensional cases.

  • Keywords

Convection-diffusion equations, finite volume simple WENO scheme, high accuracy, SSP Runge-Kutta method.

  • AMS Subject Headings

65M10, 78A48

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-17-882, author = {Deng , PufanPeng , GangFeng , XinlongLuo , Dongmi and Chen , Yibing}, title = {A Finite Volume Simple WENO Scheme for Convection-Diffusion Equations}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2024}, volume = {17}, number = {4}, pages = {882--903}, abstract = {

A simple weighted essentially non-oscillatory (SWENO) scheme for solving the convection-diffusion equation is proposed in this paper, where a seventh-order SWENO method and the third-order strong stability preserving (SSP) Runge-Kutta method are adopted for discretizing the space and time, respectively. Then the hyperbolic and diffusive part can achieve the seventh- and sixth-order accuracy, respectively. The proposed method has the following advantages. Firstly, negative linear weights are avoided. Secondly, one reconstruction with one stencil is required for the computation of convective and diffusive fluxes. Finally, the new method does not require the transformation while the diffusion coefficients are degenerate. Numerical examples demonstrate that the new method can achieve sixth-order accuracy in the smooth region and guarantee non-oscillatory properties for the discontinuous problems for one- and two-dimensional cases.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2024-0003 }, url = {http://global-sci.org/intro/article_detail/nmtma/23645.html} }
TY - JOUR T1 - A Finite Volume Simple WENO Scheme for Convection-Diffusion Equations AU - Deng , Pufan AU - Peng , Gang AU - Feng , Xinlong AU - Luo , Dongmi AU - Chen , Yibing JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 882 EP - 903 PY - 2024 DA - 2024/12 SN - 17 DO - http://doi.org/10.4208/nmtma.OA-2024-0003 UR - https://global-sci.org/intro/article_detail/nmtma/23645.html KW - Convection-diffusion equations, finite volume simple WENO scheme, high accuracy, SSP Runge-Kutta method. AB -

A simple weighted essentially non-oscillatory (SWENO) scheme for solving the convection-diffusion equation is proposed in this paper, where a seventh-order SWENO method and the third-order strong stability preserving (SSP) Runge-Kutta method are adopted for discretizing the space and time, respectively. Then the hyperbolic and diffusive part can achieve the seventh- and sixth-order accuracy, respectively. The proposed method has the following advantages. Firstly, negative linear weights are avoided. Secondly, one reconstruction with one stencil is required for the computation of convective and diffusive fluxes. Finally, the new method does not require the transformation while the diffusion coefficients are degenerate. Numerical examples demonstrate that the new method can achieve sixth-order accuracy in the smooth region and guarantee non-oscillatory properties for the discontinuous problems for one- and two-dimensional cases.

Deng , PufanPeng , GangFeng , XinlongLuo , Dongmi and Chen , Yibing. (2024). A Finite Volume Simple WENO Scheme for Convection-Diffusion Equations. Numerical Mathematics: Theory, Methods and Applications. 17 (4). 882-903. doi:10.4208/nmtma.OA-2024-0003
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