Abstract:The performance improvement of the WENO-Z+ scheme depends on the role of the additional term, which is added to the WENO-Z weights to increase the weights of less-smooth substencils further. Since the additional term may lead to negative dissipation by over increasing the weights of less-smooth substencils in smooth regions, the coefficient \lambda is set to control the role of this term and needs to be carefully determined. In this paper, the defects of the method the WENO-Z+ scheme adopts to determine the value of \lambda are pointed out: It can neither fully utilize the potential of the scheme nor effectively avoid negative dissipation. We propose that to take the full role of the additional term in reducing numerical dissipation and improving resolution ability, the value of \lambda should change with the local data of the flow field. Based on this idea, we design a new calculation formula for \lambda , which can adjust the role of the additional term adaptively: Weaken the role of the additional term only where the weights of less-smooth substencils are easy to be excessively increased. The new scheme employing the new \lambda formula is named WENO-Z++, and its numerical performance is systematically analyzed. Theoretical analysis indicates that the new scheme maintains essentially non-oscillatory (ENO) property and has lower numerical dissipation at discontinuities. The investigation of approximate dispersion relation (ADR) shows that the new scheme effectively avoids the negative dissipation caused by excessive increase of the weights of less-smoothed substencils, and its spectral properties are significantly improved. The parameters set that allow the new scheme keeping the optimal order of accuracy at extreme points is deduced. A series of numerical experiments for solving the Euler equations show that both the shock-capturing ability and resolution for complex flow structure of the new scheme are significantly better than those of the original WENO-Z+ scheme.