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.
After revision: We propose a new family of mapped WENO schemes by using 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.