Before revision: We propose  several adaptive control functions and a smoothing approximation of the signum function , and apply them to design mapping functions used to construct new mapped weighted essentially non-oscillatory (WENO) schemes. Our method extends the formulation of the mapping functions used in the WENO-IM(k,A) methods developed in the last years. The new method admits a more extensive permitted range of the parameters in the mapping functions , that it enjoys some mathematical advantages in achieving optimal convergence rates .
After revision: We propose several adaptive control functions and a smoothing approximation of the signum function . The proposed schemes introduce the adaptivity and admit an extensive permitted range of the parameters in the mapping functions , that it enjoys some mathematical advantages in achieving optimal convergence rates .