A New Mapped Weighted Essentially Non-oscillatory Method Using Rational Mapping Function

In this paper, we develop a new mapped weighted essentially non-oscillatory method for hyperbolic conservation laws by proposing a new family of mapping functions, which includes the mapping function in Henrick et al. (J Comput Phys 207:542–567, 2005) as one of its members. The new family of mapping functions has three parameters, namely m, n and k. When it is applied to the \((2r-1)\)-th-order WENO-JS methods, \(r=3\), 4, 5 and 6, it is well defined and monotonically increasing with proper choice of m, n and k. Furthermore, it can achieve the optimal order of accuracy at or near critical points in smooth regions with proper choices of n. The new family of mapping functions uses rational functions, thus it is a family of smooth functions (Note that, the mapping function in Feng et al. (J Sci Comput 51:449–473, 2012) is only piecewise continuous). Among the new family of mapping functions, the new mapping function with \((m,n,k)=(2,6,0)\) has the best performance (i.e., no matter for short time or for long time simulations, it provides the more accurate numerical solutions).