TY - JOUR
T1 - An Efficient Mapped WENO Scheme Using Approximate Constant Mapping
AU - Li , Ruo
AU - Zhong , Wei
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 1
SP - 1
EP - 41
PY - 2022
DA - 2022/02
SN - 15
DO - http://doi.org/10.4208/nmtma.OA-2021-0074
UR - https://global-sci.org/intro/article_detail/nmtma/20219.html
KW - WENO, approximate constant mapping, hyperbolic problems, high resolution, low computational cost.
AB - <p style="text-align: justify;">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
blast waves 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.</p>