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The paper analyses by Taylor series the several fifth-order of accuracy schemes for hyperbolic conservation laws: the classical WENOJS scheme [G.S. Jiang, and C.W.Shu. Efficient implementation of weighted ENO schemes. J. Comput. Phys., 126:202-228, 1996], the WENOM scheme [A.K. Henrick, T.D. Aslam, and J.M. powers. Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points. J. Comput. Phys. 207:542-567, 2005], the WENOZ scheme [R. Borges, M. Carmona, B. Costa, and W.S. Don. An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. J. Comput. Phys. 227:3191-3211, 2008] and the scheme, called WENOε here, [F. Aràndiga, A. Baeza, A.M. Belda, and P. Mulet. Analysis of WENO schemes for full and global accuracy. SIAM J. Numer. Anal., 49:893-915, 2011]. The order of weights of these four schemes agreed to the optimal weights is presented in detail. Then three prerequisites are developed if one intends to improve the WENOJS scheme: the scheme arrives the 5th-order at critical points; the weights of scheme approximate the optimal weights with high-order accuracy when solution is smooth; the scheme shouldn't introduce much oscillations intuitively in the vicinity of discontinuities. According to the prerequisites above, a new WENO scheme (MWENOZ) is devised which is similar to the WENOZ scheme. Finally, the method designed here is demonstrated robustly by applying it to 1D and 2D numerical simulations and its advantage compared with the WENOZ scheme seems more striking in 2D problems.

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