Volume 14, Issue 5
A New Fifth-Order Finite Volume Central WENO Scheme for Hyperbolic Conservation Laws on Staggered Meshes

Adv. Appl. Math. Mech., 14 (2022), pp. 1059-1086.

Published online: 2022-06

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• Abstract

In this paper, a new fifth-order finite volume central weighted essentially non-oscillatory (CWENO) scheme is proposed for solving hyperbolic conservation laws on staggered meshes. The high-order spatial reconstruction procedure using a convex combination of a fourth degree polynomial with two linear polynomials (in one dimension) or four linear polynomials (in two dimensions) in a traditional WENO fashion and a time discretization method using the natural continuous extension (NCE) of the Runge-Kutta method are applied to design this new fifth-order CWENO scheme. This new finite volume CWENO scheme uses the information defined on the same largest spatial stencil as that of the same order classical CWENO schemes [37, 46] with the application of smaller number of unequal-sized spatial stencils. Since the new nonlinear weights are adopted, the new finite volume CWENO scheme could obtain the same order of accuracy and get smaller truncation errors in $L^1$ and $L^∞$ norms in smooth regions, and control the spurious oscillations near strong shocks or contact discontinuities. The new CWENO scheme has advantages over the classical CWENO schemes [37, 46] on staggered meshes in its simplicity and easy extension to multi-dimensions. Some one-dimensional and two-dimensional benchmark numerical examples are provided to illustrate the good performance of this new fifth-order finite volume CWENO scheme.

• Keywords

Finite volume scheme, central WENO scheme, NCE of Runge-Kutta method, staggered mesh.

65M60, 35L65

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@Article{AAMM-14-1059, author = {Cui , ShengzhuTao , Zhanjing and Zhu , Jun}, title = {A New Fifth-Order Finite Volume Central WENO Scheme for Hyperbolic Conservation Laws on Staggered Meshes}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2022}, volume = {14}, number = {5}, pages = {1059--1086}, abstract = {

In this paper, a new fifth-order finite volume central weighted essentially non-oscillatory (CWENO) scheme is proposed for solving hyperbolic conservation laws on staggered meshes. The high-order spatial reconstruction procedure using a convex combination of a fourth degree polynomial with two linear polynomials (in one dimension) or four linear polynomials (in two dimensions) in a traditional WENO fashion and a time discretization method using the natural continuous extension (NCE) of the Runge-Kutta method are applied to design this new fifth-order CWENO scheme. This new finite volume CWENO scheme uses the information defined on the same largest spatial stencil as that of the same order classical CWENO schemes [37, 46] with the application of smaller number of unequal-sized spatial stencils. Since the new nonlinear weights are adopted, the new finite volume CWENO scheme could obtain the same order of accuracy and get smaller truncation errors in $L^1$ and $L^∞$ norms in smooth regions, and control the spurious oscillations near strong shocks or contact discontinuities. The new CWENO scheme has advantages over the classical CWENO schemes [37, 46] on staggered meshes in its simplicity and easy extension to multi-dimensions. Some one-dimensional and two-dimensional benchmark numerical examples are provided to illustrate the good performance of this new fifth-order finite volume CWENO scheme.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2021-0095}, url = {http://global-sci.org/intro/article_detail/aamm/20552.html} }
TY - JOUR T1 - A New Fifth-Order Finite Volume Central WENO Scheme for Hyperbolic Conservation Laws on Staggered Meshes AU - Cui , Shengzhu AU - Tao , Zhanjing AU - Zhu , Jun JO - Advances in Applied Mathematics and Mechanics VL - 5 SP - 1059 EP - 1086 PY - 2022 DA - 2022/06 SN - 14 DO - http://doi.org/10.4208/aamm.OA-2021-0095 UR - https://global-sci.org/intro/article_detail/aamm/20552.html KW - Finite volume scheme, central WENO scheme, NCE of Runge-Kutta method, staggered mesh. AB -

In this paper, a new fifth-order finite volume central weighted essentially non-oscillatory (CWENO) scheme is proposed for solving hyperbolic conservation laws on staggered meshes. The high-order spatial reconstruction procedure using a convex combination of a fourth degree polynomial with two linear polynomials (in one dimension) or four linear polynomials (in two dimensions) in a traditional WENO fashion and a time discretization method using the natural continuous extension (NCE) of the Runge-Kutta method are applied to design this new fifth-order CWENO scheme. This new finite volume CWENO scheme uses the information defined on the same largest spatial stencil as that of the same order classical CWENO schemes [37, 46] with the application of smaller number of unequal-sized spatial stencils. Since the new nonlinear weights are adopted, the new finite volume CWENO scheme could obtain the same order of accuracy and get smaller truncation errors in $L^1$ and $L^∞$ norms in smooth regions, and control the spurious oscillations near strong shocks or contact discontinuities. The new CWENO scheme has advantages over the classical CWENO schemes [37, 46] on staggered meshes in its simplicity and easy extension to multi-dimensions. Some one-dimensional and two-dimensional benchmark numerical examples are provided to illustrate the good performance of this new fifth-order finite volume CWENO scheme.

Shengzhu Cui, Zhanjing Tao & Jun Zhu. (2022). A New Fifth-Order Finite Volume Central WENO Scheme for Hyperbolic Conservation Laws on Staggered Meshes. Advances in Applied Mathematics and Mechanics. 14 (5). 1059-1086. doi:10.4208/aamm.OA-2021-0095
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