Volume 13, Issue 1
A New Fifth-Order Finite Difference WENO Scheme for Dam-Break Simulations

Xiaogang Li, Guodong LiYongbin Ge

Adv. Appl. Math. Mech., 13 (2021), pp. 58-82.

Published online: 2020-10

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  • Abstract

In this paper, a fifth-order weighted essentially nonoscillatory scheme is presented for simulating dam-break flows in a finite difference framework. The new scheme is a convex combination of two quadratic polynomials with a fourth-degree polynomial in a classical WENO fashion. The distinguishing feature of the present method is that the same five-point information is used but smaller absolute truncation errors and the same accuracy order in the smooth region are obtained. The new nonlinear weights are presented by Taylor expansion of the smoothness indicators of the small stencils to sustain the optimal fifth-order accuracy. The linear advection equation, nonlinear scalar Burgers equation, and one- and two-dimensional Euler equations are used to validate the high-order accuracy and excellent resolution of the presented method. Finally, one- and two-dimensional Saint-Venant equations are tested by using the new fifth-order scheme to simulate a dam-break flow.

  • Keywords

WENO scheme, smoothness indicators, shallow water equation, hyperbolic conservation laws.

  • AMS Subject Headings

65M60, 35L65

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-13-58, author = {Li , Xiaogang and Li , Guodong and Ge , Yongbin}, title = {A New Fifth-Order Finite Difference WENO Scheme for Dam-Break Simulations}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2020}, volume = {13}, number = {1}, pages = {58--82}, abstract = {

In this paper, a fifth-order weighted essentially nonoscillatory scheme is presented for simulating dam-break flows in a finite difference framework. The new scheme is a convex combination of two quadratic polynomials with a fourth-degree polynomial in a classical WENO fashion. The distinguishing feature of the present method is that the same five-point information is used but smaller absolute truncation errors and the same accuracy order in the smooth region are obtained. The new nonlinear weights are presented by Taylor expansion of the smoothness indicators of the small stencils to sustain the optimal fifth-order accuracy. The linear advection equation, nonlinear scalar Burgers equation, and one- and two-dimensional Euler equations are used to validate the high-order accuracy and excellent resolution of the presented method. Finally, one- and two-dimensional Saint-Venant equations are tested by using the new fifth-order scheme to simulate a dam-break flow.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2019-0155}, url = {http://global-sci.org/intro/article_detail/aamm/18340.html} }
TY - JOUR T1 - A New Fifth-Order Finite Difference WENO Scheme for Dam-Break Simulations AU - Li , Xiaogang AU - Li , Guodong AU - Ge , Yongbin JO - Advances in Applied Mathematics and Mechanics VL - 1 SP - 58 EP - 82 PY - 2020 DA - 2020/10 SN - 13 DO - http://doi.org/10.4208/aamm.OA-2019-0155 UR - https://global-sci.org/intro/article_detail/aamm/18340.html KW - WENO scheme, smoothness indicators, shallow water equation, hyperbolic conservation laws. AB -

In this paper, a fifth-order weighted essentially nonoscillatory scheme is presented for simulating dam-break flows in a finite difference framework. The new scheme is a convex combination of two quadratic polynomials with a fourth-degree polynomial in a classical WENO fashion. The distinguishing feature of the present method is that the same five-point information is used but smaller absolute truncation errors and the same accuracy order in the smooth region are obtained. The new nonlinear weights are presented by Taylor expansion of the smoothness indicators of the small stencils to sustain the optimal fifth-order accuracy. The linear advection equation, nonlinear scalar Burgers equation, and one- and two-dimensional Euler equations are used to validate the high-order accuracy and excellent resolution of the presented method. Finally, one- and two-dimensional Saint-Venant equations are tested by using the new fifth-order scheme to simulate a dam-break flow.

Xiaogang Li, Guodong Li & Yongbin Ge. (2020). A New Fifth-Order Finite Difference WENO Scheme for Dam-Break Simulations. Advances in Applied Mathematics and Mechanics. 13 (1). 58-82. doi:10.4208/aamm.OA-2019-0155
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