Experimental Convergence Rate Study for Three Shock-Capturing Schemes and Development of Highly Accurate Combined Schemes

TL;DR

This paper investigates the convergence rates of three shock-capturing schemes for hyperbolic conservation laws, revealing their accuracy limitations near shocks and proposing new combined schemes that maintain high-order accuracy and non-oscillatory behavior.

Contribution

The study provides a comprehensive experimental convergence analysis of three schemes and introduces two novel combined schemes that preserve high-order accuracy near shocks.

Findings

CU and A-WENO schemes' $W^{-1,1}$ convergence reduces to first order after shocks.

RBM scheme maintains second-order accuracy even near shocks.

New combined schemes achieve high-order accuracy in smooth regions and are non-oscillatory near shocks.

Abstract

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 $W^{-1,1}$ 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 $W^{-1,1}$ 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.