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.
Conservative approximation-based feedforward neural network for WENO schemes
Graduate School of Artificial Intelligence \& POSTECH MINDS (Mathematical Institute for Data Science), Pohang University of Science；Department of Mathematics \& POSTECH MINDS (Mathematical Institute for Data Science), Pohang University of Science
A Hermite WENO reconstruction for fourth order temporal accurate schemes based on the GRP solver for hyperbolic conservation laws
School of Mathematical Sciences, Beijing Normal University（北京师范大学数学科学学院）；Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics（应用物理与计算力学研究所计算物理实验室）
Numerical Study of Dissipative Weak Solutions for the Euler Equations of Gas Dynamics
RWTH Aachen University（亚琛工业大学）；University of Pretoria（比勒陀利亚大学）；Southern University of Science and Technology（南方科技大学）；Johannes Gutenberg-University Mainz（美因茨约翰内斯古腾堡大学）
