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.

翻译：我们研究了三种用于双曲守恒律系统的激波捕捉格式的实验收敛率：二阶中心迎风（CU）格式、三阶Rusanov-Burstein-Mirin（RBM）格式和五阶交替加权本质无振荡（A-WENO）格式。我们采用三种嵌入网格来定义实验点态、积分和$W^{-1,1}$收敛率。将所研究格式应用于浅水方程，并对其进行了全面的数值收敛性研究。我们验证了在光滑解区域这些格式能达到其理论精度阶数，但在激波形成后，部分计算解受激波传播影响，点态和积分收敛率均有所下降。此外，依靠非线性稳定机制的CU和A-WENO格式的$W^{-1,1}$收敛率降至一阶，而采用线性稳定化的RBM格式则清晰保持二阶精度。最后，基于实验收敛率研究，我们开发了两种以RBM格式为基础、分别与CU或A-WENO格式相结合的新组合格式。所得组合格式在光滑区域能达到与RBM格式相同的高阶精度，同时在激波附近保持无振荡特性。