An Efficient Mapped WENO Scheme Using Approximate Constant Mapping

Abstract

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.

An Extension of the Order-Preserving Mapping to the WENO-Z-Type Schemes

“…In Tables 9-12, the CPU time-consuming of the WENO-JS, WENO-M, WENO-Z and MOP-GMWENO-Z schemes and the extra cost (in brackets) of the WENO-M, WENO-Z and MOP-GMWENO-Z schemes are presented. From Tables 9-12, we can find that: (1) for each example with different mesh sizes, the CPU time-consuming in the three tests has a certain degree of fluctuation; (2) the most efficient scheme is WENO-Z and the most inefficient one is WENO-M whose extra cost compared to WENO-JS is about 25% to 30%, and these results are consistent with those obtained in [33,44]; (3) as expected, the computational cost of MOP-GMWENO-Z has increased due to the additional operations for the order-preserving implementation as shown in Algorithm 1; (4) however, the extra costs of MOP-GMWENO-Z for all tests are no more than 10% on average, and they are far less than those obtained by the WENO-M scheme.…”

“…It was reported [40] that there are three versions of the odd-order WENO methods with r ≥ 3 when applied to hyperbolic systems. They are the classical WENO schemes (e.g., WENO-JS [22], WENO-ZS [41], WENO-NIS [13]), the mapped WENO schemes (e.g., WENO-M [29], WENO-PMk [31], WENO-MAIMi [32], WENO-ACM [44]) and the WENO-Z-type schemes (e.g., WENO-Z [33], WENO-Z+ [10], WENO-NIP [21]). As the version of the mapped WENO schemes has been discussed carefully in our previous work [36], we mainly describe the other two versions in this section.…”

“…It is well known [29,30,32,33,44] that the WENO-JS scheme can not achieve the designed convergence order at critical points, and there are usually two ways to address this issue: one is to introduce a mapping function [29-32, 37, 38, 44], the other is to introduce a GSI [10,14,20,21,33,39]. Recently, the long-run simulations of the mapped WENO methods have been widely concerned [30][31][32]34,36,37,43].…”

A smoothness indicators free third-order weighted ENO scheme for gas-dynamic Euler equations

“…From equation (15), one can see that the weight function in WENO-Z scheme does not satisfy the sufficient conditions given by Yamaleev et al However, the numerical results show that the WENO-Z can achieve the optimal accuracy at the 0th order critical point.…”

“…And they developed a mapping function, which resulted in a fifth-order accuracy WENO-M method that can achieve the best at the first-order critical point. Since then, Feng [9,10], Wang [11], Vevek [12], Hong [13], Hu [14], Li [15,16],…”