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Experimental Convergence Rate Study for Three Shock-Capturing Schemes and Development of Highly Accurate Combined Schemes
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.Comment: 33 page