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We study experimental convergence rates of three shock-capturing schemes forhyperbolic systems of conservation laws: the second-order central-upwind (CU)scheme, the third-order Rusanov-Burstein-Mirin (RBM), and the fifth-orderalternative weighted essentially non-oscillatory (A-WENO) scheme. We use threeimbedded grids to define the experimental pointwise, integral, and $W^{-1,1}$convergence rates. We apply the studied schemes to the shallow water equationsand conduct their comprehensive numerical convergence study. We verify thatwhile the studied schemes achieve their formal orders of accuracy on smoothsolutions, after the shock formation, a part of the computed solutions isaffected by shock propagation and both the pointwise and integral convergencerates reduce there. Moreover, while the $W^{-1,1}$ convergence rates for the CUand A-WENO schemes, which rely on nonlinear stabilization mechanisms, reduce tothe first order, the RBM scheme, which utilizes a linear stabilization, isclearly second-order accurate. Finally, relying on the conducted experimentalconvergence rate study, we develop two new combined schemes based on the RBMand either the CU or A-WENO scheme. The obtained combined schemes can achievethe same high-order of accuracy as the RBM scheme in the smooth areas whilebeing non-oscillatory near the shocks.

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