Mapped CWENO scheme for hyperbolic conservation laws

In this paper, we analysis the Central Weighted Essentially Non-Oscillatory (CWENO) scheme for hyperbolic conservation laws (Qiu and Shu, 2002). The CWENO procedure contains three major steps: the reconstruction of ūi+1∕2n from ūin, the reconstruction of uin from ūin, and the approximation of ∫tntn+1f(u(xi,t))dt using the Natural Continuous Extension (NCE) of Runge-Kutta method. We found that both reconstructions lost the designed optimal order of accuracy near critical points. This loss of accuracy can be recovered by using a proper mapping function. Furthermore, numerical tests demonstrate that, in practice, it is enough for the mapping technique to be implemented only in the first reconstruction. This new mapped central WENO method is not only more accurate than the one in Qiu and Shu (2002), but also more efficient.