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Summary: This paper presents a new WENO‐Z scheme (WENO‐MZ) that incorporates a mapping function to enhance the weights of the less smooth sub‐stencils. The mapping function uses an innovative approach to modify the weight ratio of the less smooth sub‐stencil to the smooth stencil. In addition, we present the WENO‐MD scheme, which is a further development of the WENO‐MZ scheme that incorporates a modifier function. The WENO‐MD scheme shows improvements over the WENO‐MZ scheme by achieving an improved optimal order at critical points in higher orders and by increasing the proportion of less smooth sub‐stencils. Theoretical and numerical experiments have shown that the newly developed methods have improved shock capture capabilities and resolution compared to WENO‐JS, WENO‐Z, WENO‐M, WENO‐D, and WENO‐AIM, and also lead to significant computational time savings compared to WENO‐M and WENO‐AIM. [ABSTRACT FROM AUTHOR]

In this article, we improved the nonlinear weight of the classical third‐order WENO scheme by using the limiter of the MUSCL scheme and got a new WENO scheme, the WENO‐F scheme, and developed three new nonlinear multiple‐parameter weight functions by improving the nonlinear weight of WENO‐JS, WENO‐Z, and WENO‐Z+ schemes. After selecting appropriate parameters of these new weight functions, three new high‐resolution schemes are comparing with three kinds of classical WENO schemes. Theoretical analysis and numerical results show that the new scheme with these multiparameter functional weights can achieve third‐order convergence in smooth regions. And the new scheme has lower dissipation and better resolution than the classical third‐order WENO schemes for smooth and discontinuous solutions. [ABSTRACT FROM AUTHOR]