Well-balanced path-conservative central-upwind schemes based on flux globalization
In this paper, we introduce a new approach for constructing robust well-balanced (WB) finite-volume methods for nonconservative one-dimensional hyperbolic systems of nonlinear partial differential equations. The WB property, namely, the ability of the scheme to exactly preserve physically relevant steady-state solutions is enforced using a flux globalization approach according to which a studied system is rewritten in an equivalent quasi-conservative form with global fluxes. To this end, one needs to incorporate nonconservative product terms into the global fluxes. The resulting system can then be solved using a Riemann-problem-solver-free central-upwind (CU) scheme. However, a straightforward integration of the nonconservative terms would result in a scheme capable of exactly preserving very simple smooth steady states only and failing to preserve discontinuous steady states naturally arising in the nonconservative models. In order to ameliorate the flux globalization based CU scheme, we evaluate the integrals of the nonconservative product terms using the technique introduced in , where a path-conservative central-upwind scheme (PCCU) was introduced. This results in a new flux globalization based WB PCCU scheme, which is much more accurate and robust than both the original PCCU scheme and the straightforward flux globalization based CU scheme. This is illustrated using two nonconservative systems: a system describing fluid flows in nozzles with variable cross-sections and the two-layer shallow water equations. We demonstrate superiority of the proposed flux globalization based WB PCCU scheme on a number of challenging examples.
First ; Corresponding
National Natural Science Foundation of China;National Natural Science Foundation of China;National Natural Science Foundation of China;Guangdong Provincial Key Laboratory Of Computational Science And Material Design[2019B030301001];
|WOS Research Area|
Computer Science ; Physics
Computer Science, Interdisciplinary Applications ; Physics, Mathematical
|WOS Accession No|
|ESI Research Field|
Cited Times [WOS]:1
|Document Type||Journal Article|
|Department||Department of Mathematics|
1.Department of Mathematics,SUSTech International Center for Mathematics and Guangdong Provincial Key Laboratory of Computational Science and Material Design,Southern University of Science and Technology,Shenzhen,518055,China
2.Department of Mathematics,Southern University of Science and Technology,Shenzhen,518055,China
3.Institute of Mathematics,University of Zürich,Zürich,8057,Switzerland
|First Author Affilication||Department of Mathematics; SUSTech International Center for Mathematics|
|Corresponding Author Affilication||Department of Mathematics|
|First Author's First Affilication||Department of Mathematics; SUSTech International Center for Mathematics|
Kurganov，Alexander,Liu，Yongle,Xin，Ruixiao. Well-balanced path-conservative central-upwind schemes based on flux globalization[J]. JOURNAL OF COMPUTATIONAL PHYSICS,2023,474.
Kurganov，Alexander,Liu，Yongle,&Xin，Ruixiao.(2023).Well-balanced path-conservative central-upwind schemes based on flux globalization.JOURNAL OF COMPUTATIONAL PHYSICS,474.
Kurganov，Alexander,et al."Well-balanced path-conservative central-upwind schemes based on flux globalization".JOURNAL OF COMPUTATIONAL PHYSICS 474(2023).
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