A Modified Adaptive Improved Mapped WENO Method

We propose a new family of mapped WENO schemes by using several adaptive control functions and smoothing approximation the signum function. The proposed introduce adaptivity admit an extensive permitted range parameters in mapping functions. Consequently, they have capacity to achieve optimal convergence rates, even near critical points. Particularly, one these with fine-tuned illustrates significant advantage when solving problems discontinuities. It produces numerical solutions high resolution without generating spurious oscillations, especially for long output times.