Experimental convergence rate study for three shock‐capturing schemes and development of highly accurate combined schemes

Experimental convergence rate study for three shock‐capturing schemes and development of highly accurate combined schemes Full article

Journal

Numerical Methods for Partial Differential Equations
ISSN: 0749-159X

Output data

Year: 2023, Volume: 39, Number: 6, Pages: 4317-4346 Pages count : 30 DOI: 10.1002/num.23053

Tags

combinedschemes,finite-differenceschemes,finite-volume methods, integral convergence, orderreduction behind the shocks, pointwise convergence

Authors

Shaoshuai Chu ¹ , О. А. Ковыркина ² , Alexander Kurganov ³ , В. В. Остапенко ²

Affiliations

1	Department of Mathematics, Southern University of Science and Technology, Shenzhen, China
2	Lavrentyev Institute of Hydrodynamics of SB RAS, Novosibirsk State University, Novosibirsk, Russia
3	Department of Mathematics, Shenzhen International Center for Mathematics and Guangdong Provincial Key Laboratory of Computational Science and Material Design, Southern University of Science and Technology, Shenzhen, China

Funding (2)

1	Российский научный фонд	22-11-00060
2	Российский фонд фундаментальных исследований	21-51-53012 ГФЕН_а

We study experimental convergence rates of three shock-capturing schemes for hyperbolic systems of conservation laws: the second-order central-upwind (CU) scheme, the third-order Rusanov-Burstein-Mirin (RBM), and the fifth-order alternative weighted essentially non-oscillatory (A-WENO) scheme. We use three imbedded grids to define the experimental pointwise, integral, and convergence rates. We apply the studied schemes to the shallow water equations and conduct their comprehensive numerical convergence study. We verify that while the studied schemes achieve their formal orders of accuracy on smooth solutions, after the shock formation, a part of the computed solutions is affected by shock propagation and both the pointwise and integral convergence rates reduce there. Moreover, while the convergence rates for the CU and A-WENO schemes, which rely on nonlinear stabilization mechanisms, reduce to the first order, the RBM scheme, which utilizes a linear stabilization, is clearly second-order accurate. Finally, relying on the conducted experimental convergence rate study, we develop two new combined schemes based on the RBM and either the CU or A-WENO scheme. The obtained combined schemes can achieve the same high order of accuracy as the RBM scheme in the smooth areas while being non-oscillatory near the shocks

Shaoshuai Chu , О. А. Ковыркина , Alexander Kurganov , В. В. Остапенко
Experimental convergence rate study for three shock‐capturing schemes and development of highly accurate combined schemes
Numerical Methods for Partial Differential Equations. 2023. V.39. N6. P.4317-4346. DOI: 10.1002/num.23053 WOS Scopus РИНЦ OpenAlex

Web of science:	WOS:001007616500001
Scopus:	2-s2.0-85161911466
Elibrary:	60042950
OpenAlex:	W4380606450

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