Before revision:  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 . Improved numerical results are obtained for one-dimensional linear advection cases, especially over long output times .
After revision:  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 , even near critical points. Particularly, one of these new schemes with fine-tuned parameters illustrates a significant advantage when solving problems with discontinuities. It produces numerical solutions with high resolution without generating spurious oscillations, especially for long output times .