  We present a novel mapping approach for WENO schemes through the use of an
approximate constant mapping function which is constructed by employing an
approximation of the classic signum function. The new approximate constant
mapping function is designed to meet the overall criteria for a proper mapping
function required in the design of the WENO-PM6 scheme. The WENO-PM6 scheme was
proposed to overcome the potential loss of accuracy of the WENO-M scheme which
was developed to recover the optimal convergence order of the WENO-JS scheme at
critical points. Our new mapped WENO scheme, denoted as WENO-ACM, maintains
almost all advantages of the WENO-PM6 scheme, including low dissipation and
high resolution, while decreases the number of mathematical operations
remarkably in every mapping process leading to a significant improvement of
efficiency. The convergence rates of the WENO-ACM scheme have been shown
through one-dimensional linear advection equation with various initial
conditions. Numerical results of one-dimensional Euler equations for the
Riemann problems, the Mach 3 shock-density wave interaction and the
Woodward-Colella interacting blastwaves are improved in comparison with the
results obtained by the WENO-JS, WENO-M and WENO-PM6 schemes. Numerical
experiments with two-dimensional problems as the 2D Riemann problem, the
shock-vortex interaction, the 2D explosion problem, the double Mach reflection
and the forward-facing step problem modeled via the two dimensional Euler
equations have been conducted to demonstrate the high resolution and the
effectiveness of the WENO-ACM scheme. The WENO-ACM scheme provides
significantly better resolution than the WENO-M scheme and slightly better
resolution than the WENO-PM6 scheme, and compared to the WENO-M and WENO-PM6
schemes, the extra computational cost is reduced by more than 83% and 93%,
respectively.
