Well-balanced path-conservative central-upwind schemes based on flux globalization

Kurganov，Alexander1; Liu，Yongle2,3; Xin，Ruixiao2

2023-02-01

10.1016/j.jcp.2022.111773

JOURNAL OF COMPUTATIONAL PHYSICS   Impact Factor & Quartile

0021-9991

1090-2716

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 [5], 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.

Flux globalization Nozzle flow Path-conservative central-upwind schemes Two-layer shallow water equations Well-balanced schemes

SCI

English

First ; Corresponding

National Natural Science Foundation of China[11771201];National Natural Science Foundation of China[12111530004];National Natural Science Foundation of China[12171226];Guangdong Provincial Key Laboratory Of Computational Science And Material Design[2019B030301001];

Computer Science ; Physics

Computer Science, Interdisciplinary Applications ; Physics, Mathematical

WOS:000915940100005

ACADEMIC PRESS INC ELSEVIER SCIENCE

PHYSICS

2-s2.0-85142355478

Scopus

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

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).