An improved mapped weighted essentially non-oscillatory scheme

In this paper we develop an improved version of the mapped weighted essentially non-oscillatory (WENO) method in Henrick et al. (2005) [10] for hyperbolic partial differential equations. By rewriting and making a change to the original mapping function, a new type of mapping functions is obtained. There are two parameters, namely A and k, in the new mapping functions (see Eq. (13)). By choosing k=2 and A=1, it leads to the mapping function in Henrick et al., i.e.; the mapped WENO method by Henrick et al. actually belongs to the family of our improved mapped WENO schemes. Furthermore, we show that, when the new mapping function is applied to any (2r-1)th order WENO scheme for proper choice of k, it can achieve the optimal order of accuracy near critical points. Note that, if only one mapping is used, the mapped WENO method by Henrick et al., whose order is higher than five, can not achieve the optimal order of accuracy in some cases. Through extensive numerical tests, we draw the conclusion that, the mapping function proposed by Henrick et al. is not the best choice for the parameter A. A new mapping function is then selected and provides an improved mapped WENO method with less dissipation and higher resolution. (C) 2014 Elsevier Inc. All rights reserved.