TY - JOUR

T1 - Experimental convergence rate study for three shock-capturing schemes and development of highly accurate combined schemes

AU - Chu, Shaoshuai

AU - Kovyrkina, Olyana A.

AU - Kurganov, Alexander

AU - Ostapenko, Vladimir V.

N1 - The reported study was funded in part by RFBR and NSFC, project numbers 21‐51‐53012 (RFBR) and 12111530004 (NSFC). The work of Olyana A. Kovyrkina and Vladimir V. Ostapenko on the development of a methodology for assessing the accuracy of shock‐capturing schemes was supported by the Russian Science Foundation (project number 22‐11‐00060). The work of Alexander Kurganov was supported in part by NSFC grant 12171226 and by the fund of the Guangdong Provincial Key Laboratory of Computational Science and Material Design (No. 2019B030301001).

PY - 2023/11

Y1 - 2023/11

N2 - 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 (Formula presented.) 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 (Formula presented.) 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.

AB - 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 (Formula presented.) 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 (Formula presented.) 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.

KW - combined schemes

KW - finite-difference schemes

KW - finite-volume methods

KW - integral convergence

KW - order reduction behind the shocks

KW - pointwise convergence

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85161911466&origin=inward&txGid=65a6b711273ee328e8c72952d2cc349e

UR - https://www.mendeley.com/catalogue/bc906160-382e-340f-84c6-375d6f7e09fc/

U2 - 10.1002/num.23053

DO - 10.1002/num.23053

M3 - Article

VL - 39

SP - 4317

EP - 4346

JO - Numerical Methods for Partial Differential Equations

JF - Numerical Methods for Partial Differential Equations

SN - 0749-159X

IS - 6

ER -