Volume 27, Issue 5
Implementation of Finite Difference Weighted Compact Nonlinear Schemes with the Two-Stage Fourth-Order Accurate Temporal Discretization

Zhiwei He, Fujie Gao, Baolin Tian & Jiequan Li

Commun. Comput. Phys., 27 (2020), pp. 1470-1484.

Published online: 2020-03

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  • Abstract

In this paper, we present a new two-stage fourth-order finite difference weighted compact nonlinear scheme (WCNS) for hyperbolic conservation laws with special application to compressible Euler equations. To construct this algorithm, apart from the traditional WCNS for the spatial derivative, it was necessary to first construct a linear compact/explicit scheme utilizing time derivative of flux at midpoints, which,in turn, was solved by a generalized Riemann solver. Combining these two schemes,the fourth-order time accuracy was achieved using only the two-stage time-stepping technique. The final algorithm was numerically tested for various one-dimensional and two-dimensional cases. The results demonstrated that the proposed algorithm had an essentially similar performance as that based on the fourth-order Runge-Kutta method, while it required 25 percent less computational cost for one-dimensional cases, which is expected to decline further for multidimensional cases.

  • Keywords

Hyperbolic conservation laws, finite difference method, Lax-Wendroff type time dis-cretization, WCNS.

  • AMS Subject Headings

65M06, 65M20, 35L65, 35L04

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

he zhiwei@iapcm.ac.cn (Zhiwei He)

gao fujie@iapcm.ac.cn (Fujie Gao)

tian baolin@iapcm.ac.cn (Baolin Tian)

li jiequan@iapcm.ac.cn (Jiequan Li)

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@Article{CiCP-27-1470, author = {He , Zhiwei and Gao , Fujie and Tian , Baolin and Li , Jiequan }, title = {Implementation of Finite Difference Weighted Compact Nonlinear Schemes with the Two-Stage Fourth-Order Accurate Temporal Discretization}, journal = {Communications in Computational Physics}, year = {2020}, volume = {27}, number = {5}, pages = {1470--1484}, abstract = {

In this paper, we present a new two-stage fourth-order finite difference weighted compact nonlinear scheme (WCNS) for hyperbolic conservation laws with special application to compressible Euler equations. To construct this algorithm, apart from the traditional WCNS for the spatial derivative, it was necessary to first construct a linear compact/explicit scheme utilizing time derivative of flux at midpoints, which,in turn, was solved by a generalized Riemann solver. Combining these two schemes,the fourth-order time accuracy was achieved using only the two-stage time-stepping technique. The final algorithm was numerically tested for various one-dimensional and two-dimensional cases. The results demonstrated that the proposed algorithm had an essentially similar performance as that based on the fourth-order Runge-Kutta method, while it required 25 percent less computational cost for one-dimensional cases, which is expected to decline further for multidimensional cases.

}, issn = {1991-7120}, doi = {https://doi.org/ 10.4208/cicp.OA-2019-0029}, url = {http://global-sci.org/intro/article_detail/cicp/15765.html} }
TY - JOUR T1 - Implementation of Finite Difference Weighted Compact Nonlinear Schemes with the Two-Stage Fourth-Order Accurate Temporal Discretization AU - He , Zhiwei AU - Gao , Fujie AU - Tian , Baolin AU - Li , Jiequan JO - Communications in Computational Physics VL - 5 SP - 1470 EP - 1484 PY - 2020 DA - 2020/03 SN - 27 DO - http://dor.org/ 10.4208/cicp.OA-2019-0029 UR - https://global-sci.org/intro/cicp/15765.html KW - Hyperbolic conservation laws, finite difference method, Lax-Wendroff type time dis-cretization, WCNS. AB -

In this paper, we present a new two-stage fourth-order finite difference weighted compact nonlinear scheme (WCNS) for hyperbolic conservation laws with special application to compressible Euler equations. To construct this algorithm, apart from the traditional WCNS for the spatial derivative, it was necessary to first construct a linear compact/explicit scheme utilizing time derivative of flux at midpoints, which,in turn, was solved by a generalized Riemann solver. Combining these two schemes,the fourth-order time accuracy was achieved using only the two-stage time-stepping technique. The final algorithm was numerically tested for various one-dimensional and two-dimensional cases. The results demonstrated that the proposed algorithm had an essentially similar performance as that based on the fourth-order Runge-Kutta method, while it required 25 percent less computational cost for one-dimensional cases, which is expected to decline further for multidimensional cases.

Zhiwei He , Fujie Gao , Baolin Tian & Jiequan Li. (2020). Implementation of Finite Difference Weighted Compact Nonlinear Schemes with the Two-Stage Fourth-Order Accurate Temporal Discretization. Communications in Computational Physics. 27 (5). 1470-1484. doi: 10.4208/cicp.OA-2019-0029
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