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

To improve the shock-capturing capability of the third-order WENO scheme and enhance its computational efficiency, in this paper, we design a new WENO scheme independent of the local smoothing factor, WENO-SIF. The weight function of the WENO-SIF scheme is the segmentation function of the sub-stencil, which is guaranteed to achieve the desired accuracy at higher order critical points. WENO-SIF does not need to compute the smoothing factor during the computation, which effectively reduces the computational consumption. The present WENO-SIF is compared with WENO-JS and other WENO schemes for numerical experiments at one- and two-dimensional benchmark problems with a suitable choice of λ=0.13\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda =0.13$$\end{document}. The results demonstrate that the WENO scheme can further improve the resolution of WENO-JS, achieve optimal accuracy at high-order critical points, and significantly reduce the computational consumption.


Introduction
Based on the ENO scheme (Harten 1983;Liu et al. 1994) proposed the WENO method that can maintain the ENO property in the smooth region by recombining the sub-stencils using a weighting function. Due to its high accuracy and essentially oscillatory WENO scheme, it has been widely used and vigorously developed in fluid mechanics and aerospace. Jiang and Shu (1996) proposed a technique that can measure the smoothness of subtemplates, and they gave a framework for building the WENO method with various accuracies. Since then, Communicated by José R Fernáández. the researchers have conducted tests, evaluations, and improvements on the WENO method (Balsara and Shu 2000;Qiu and Shu 2003;Rathan and Raju 2018;. Henrick et al. (2005) found that the fifth-order WENO-JS scheme suffers from accuracy degradation at the extremum. They developed a mapping function that resulted in a fifthorder accurate WENO-M method that achieves the best performance at the first-order critical point. Since then, Feng et al. (2012Feng et al. ( , 2014, Wang et al. (2016), Vevek et al. (2019), Hong et al. (2020), , Li and Zhong (2021),  and Zhu and Qiu (2021) have developed a series of mapping functions successively to develop different higher-order mapping type WENO schemes. Unlike Henrick's mapping-type WENO scheme, Borges et al. (2008) designed a new type of nonlinear weights by using a linear combination of low-order local smoothness factors to construct a global high-order smoothness factor. They developed a class of structurally simple fifth-order WENO-Z schemes. In addition, Castro et al. (2011), Don and Borges (2013), Liu et al. (2018), Wang et al. (2018), Peng et al. (2019), Baeza et al. (2019aBaeza et al. ( , b, 2020, , Semplice et al. (2016), Dumbser et al. (2017), Cravero et al. (2018), Rathan et al. (2020) and Huang and Chen (2021) have successively developed various high-order WENO-Z type schemes based on Borges'.
Compared to higher-order WENO schemes of order 5 and above, the third-order WENO scheme is more robust in capturing shocks, uses fewer grid points, and can be easily extended to unstructured grids. However, it also has shortcomings, such as higher dissipation and lower accuracy. Yamaleevn and Carpenter (2009) developed a high-resolution third-order energystable WENO scheme (ESWENO) using an amount-preserving stable global smoothness factor. Liu et al. (2017) proposed the WENO-MN scheme using the values of all three points on the global stencil of the third-order WENO scheme to calculate the local smoothness factor. Zhao (2015, 2016) and Xu and Wu (2018) developed three WENO schemes, namely WENO-N, WENO-NP, and WENO-NN, using different nonlinear combinations of local smoothness indicators and local smoothness factors.  introduced a new reference smoothness indicator to construct a third-order WENO-R scheme with low dissipation.  constructed a new global smoothness indicator by using Taylor expansions to handle local smoothness indicators and developed the WENO-ZF scheme.  designed two new global smoothness indicators by nonlinearly combining local smoothness indicators with reference values based on Lagrangian interpolation polynomials and developed the WENO-L3 and WENO-L4. Tang and Li (2021) constructed three multiparameter types of finite-volume WENO schemes by modifying the weight functions of the WENO scheme using the construction of the limiter of the MUSCL scheme. In addition, many other scholars have improved the third-order WENO scheme and developed numerous new schemes (Kumar and Kaur 2020;Kim et al. 2021).
However, various current third-order WENO schemes suffer from two shortcomings: First, they fail to achieve optimal accuracy at high-order critical points. Second, some highresolution WENO schemes suffer from a relatively complicated construction of the weight function and low computational efficiency. To address these issues, in this paper we improve the WENO scheme by constructing a weight function that does not depend on the local smoothing factor. The results of WENO-SIF show that it can achieve optimal weights at higher order critical points and has a simple and efficient structure. The present WENO-SIF scheme with λ = 0.13 has better spectral properties and higher resolution through ADR analysis (Pirozzoli 2006) and calculation of various 1D-2D problems, and its computational efficiency is much higher than other given WENO schemes.
We set the basic framework for this paper: Sect. 2 presents the structure of the classical third-order WENO scheme. In Sect. 3 we give construction hints for the new WENO scheme and determine the parameters of the weight function in the new scheme using the ADR method. In Sect. 4, we compare the performance of various WENO schemes in terms of accuracy at higher-order critical points, resolution, and computational efficiency for 1-D and 2-D Eulerian gas dynamics. We provide the conclusions of this paper in Sect. 5.

Reviews of WENO methods
In this section, we briefly introduce a series of third-order WENO schemes. Consider the following one-dimensional scalar hyperbolic conservation laws. (1) Discretizing the computational region as a uniform interval of x yields a conservative scheme of semi-differentiation of the following form: wheref j+1/2 =f + j+1/2 +f − j+1/2 is the numerical flux, which satisfies df + (u)/du ≥ 0 and df − (u)/du ≤ 0. Sincef + j+1/2 andf − j+1/2 are approximated similarly, for convenience, we will only describe the approximation off + j+1/2 . For simplicity, we will remove the superscript + of the flux.
On the 3-point stencil at the central node x j−1/2 , the numerical flux function of the thirdorder finite-difference WENO scheme is defined as: where q k is the second-order flux on the sub-stencil S k = {x j−1+k , x j+k }, k = 0, 1, which is defined by And the weights ω k will be introduced in the following subsections.

WENO-JS scheme
The nonlinear weights proposed by Jiang and Shu (1996) are where (d 0 , d 1 ) = (1/3, 2/3). is a positive number with the value 10 −6 to avoid the denominator becoming zero in the smoothing region. The local smoothness indicators β k in the stencils are calculated by where r = 2 is the number of sub-stencils. And the local smoothness indicators of the third-order WENO scheme are Taylor expansion of Eq. (7) at the point x j yields Easy to verify that where n cp is a integer that satisfies the following condition: if the function f (x) has n cp + 1-order derivatives and satisfies, f , then x 0 is said to be a n cp -order critical point of the function f (x). Substituting (9) into (5) and ignoring , we get To get high accuracy numerical solution, the nonlinear weights of the WENO scheme should be approximately equal to the linear at the smooth region. Yamaleevn and Carpenter (2009) proposed a sufficient condition for the third-order accuracy of the weights ω k It can be seen that WENO-JS will reduce the order at the high-order critical point.

WENO-Z-type scheme
To improve the accuracy of WENO-JS, Borges et al. (2008) constructed a fifth-order WENO-Z scheme capable of achieving the optimal accuracy at the first-order critical point by introducing a global smoothness indicator. Similar to the fifth-order WENO-Z scheme, the weights of the third-order WENO-Z-type scheme are calculate as following From this, the weight functions of the third-order WENO-Z scheme can be got as: In the smooth region, the Taylor expansion of Eq. (13) at the point x j gives Thus 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. Xu and Wu (2018) proposed a new WENO-Z type scheme, WENO-PZ, by slightly modifying the local smoothness indicator of the third-order WENO-Z by using Taylor expansion. And they gave the following new reference smoothness indicator To enable the WENO-PZ scheme to achieve high accuracy at the critical points, Xu et al. set the value of p to 3/4. However, since the exponents of the local-smooth factors in WENO-PZ are rational numbers, they consume a lot of computational time during the calculation, a conclusion we will give later in the numerical calculation section.

WENO-M scheme
To overcome the fifith-order WENO-JS scheme will lose accuracy at the critical points, Henrick et al. (2005) designed the following mapping function to improve the approximation Similar to the fifth-order WENO-M scheme, the weights of the third-order WENO-M-type are as follows A Taylor expansion of Eq. (17) at d k gives Thus that is, the WENO-M satisfies the sufficient condition for the third-order accuracy of the weights. Interestingly, although WENO-M can achieve theoretically the optimal accuracy, it is not computationally achievable. This conclusion we will present in the section on numerical calculations.

WENO-MN scheme
In the smooth region, the Taylor series expansions of (21) are Thus, the weights of the WENO-JS-type scheme WENO-MN satisfies Liu et al. pointed out that the weights of the WENO-MN scheme do not meet the sufficient condition near the critical point, and its numerical dissipation will be significantly smaller than that of the WENO-JS in the smooth region.

Design and properties of the new WENO scheme
In this section, we propose a new WENO scheme independent of the local smoothing factor. The specific construction process of WENO-SIF is divided into the following parts. First, we reconstruct the weight functions of WENO-JS. It may be assumed that where is a positive number to prevent the numerator and denominator from being equal to zero, and in this paper, we always take the value = 10 −40 without specifical specification. Thus, the weight functions of WENO-JS can always be rewritten in the following form Similarly, the weight functions of the WENO-Z scheme can be rewritten as In contrast, the weight functions of the WENO-MN scheme can be rewritten as Second, combining the characteristics of the above several WENO schemes, we can always construct various improved WENO schemes without local smooth factors that satisfy the third-order condition by using functions of r j . That leads to a conclusion as follows.
Theorem 1 If the weight functions ω k (r j ), k = 0, 1 satisfies the following conditions: (a) ω 0 (r j ) (or ω 1 (r j )) is monotone decrease (or increase) with r j , and satisfying ω 0 (0) = 1 Then, the resulting WENO scheme can achieve optimal accuracy at any critical point.
Proof Ignoring in Eq. (24), we get Suppose that ω k (r j ), k = 0, 1 satisfies the two conditions in the theorem, then These weight functions will satisfy the sufficient conditions for third-order accuracy at anyorder critical point.
To reduce the consumption of the weight function in the calculation process and ensure that the new WENO scheme can achieve optimal accuracy at any critical point, we construct weight functions of the following form by combining the two conditions in Theorem 1.
in which, λ is a positive number less than 1 to be determined. We refer to the new WENO scheme equipped with the weight function of (30) as the local smooth factor-free WENO scheme, or WENO-SIF for short.
It is easy to verify that WENO-SIF satisfies all the conditions of Theorem 1 for λ < 1, and we can also get the following properties.
Proposition Consider S C and S D are the sub-stencils of WENO-SIF, and S D is the sub-stencil where the solution less smooth than in S C , i.e. β D > β C . λ 1 and λ 2 are positive numbers, and satisfy λ 1 < λ 2 < 1. Then Then Thus, for arbirtrary positive numbers λ 1 and λ 2 , if λ 1 < λ 2 , there is Figure 2 depicts the spectral properties of WENO-SIF by using the approximate dispersion relation of the shock capture method (ADR) (Pirozzoli 2006). We can see that when λ → 0, the spectrum of WENO-SIF is close to the third-order linear upwind scheme. For good numerical stability and high resolution of WENO-SIF in numerical calculations, as a rule of thumb, we will always choose λ = 0.13 in this paper. We will further discuss the impact of λ on the scheme in the numerical calculations thereafter.   We also compared the dispersion and dispersion of the third-order WENO-JS, WENO-Z, WENO-M, WENO-PZ ( p = 3/4), WENO-MN, WENO-SIF, and the third-order upwind scheme, and the results shown in Fig. 3. From the figure, one can see that WENO-SIF has better spectral characteristics than other WENO schemes.

Numerical results
In this section, we will compare the performance of the present third-order WENO-SIF with WENO-JS, WENO-Z, WENO-M, WENO-PZ and WENO-MN by computing several classical problems, where the time advance is the third-order TVD Runge-Kutta method (Jiang and Shu 1996).
where C[u n ] denote the numerical flux. Without specified, we will set to be 1e−40 in the following, except for WENO-JS ( = 1e−6), and the CFL number is 0.6. In the following, CPU times for all 1-D problems are the averaged values of 100 calculations run with an Intel i3-10100 @ 3.60 GHz processor.

Convergence at higher-order critical points
In this subsection, we will compare various WENO schemes by computing the accuracy of a function at the critical points of the selected order. Given a trial function of the following form Then, the value of the kth order derivative of this function at the point x 0 = 0 is that is, x 0 = 0 is the n cp order critical point of f (x). For each n cp , we investigate the convergence of the critical point x 0 = 0 at six levels from q = 0 to 5. Where each level defined as x = 0.001/2 q . Tables 1, 2 and 3 list the L 1 , L 2 and L ∞ errors and convergence rates in the approximation of Table 1 shows all the computational errors and accuracies of various WENO schemes for the n cp = 0 critical point x c = 0. One can see that all the WENO schemes achieve third-order accuracy. The errors of WENO-Z, WENO-M, WENO-PZ, WENO-MN, and WENO-SIF are about the same and smaller than those of the WENO-JS scheme. That shows the WENO-SIF scheme can achieve optimal accuracy at the 0th-order critical point. Table 2 represents the various computational errors and accuracies at the n cp = 1 critical point. The accuracy of all known WENO schemes degrades with WENO-JS and WENO-Z reduced to first-order. And the WENO-SIF scheme provided in this paper maintains the same errors and accuracies as at the zero-order critical point. That shows the WENO-SIF method can achieve optimal accuracy at the first-order critical point. Table 3 represents the various computational errors and accuracies for n cp = 2. The results show that the accuracy of the methods WENO-JS, WENO-Z, and WENO-M decreases to the second-order at the second-order critical point, and the WENO-PZ can achieve the third-order accuracy. In contrast, the WENO-SIF scheme has the smallest various errors and maintains the theoretical third-order accuracy.

One-dimensional linear advection problems
The main work of this subsection is to examine the resolution and computational efficiency of various WENO schemes in computing the following 1D advection equation

Case 1
The initial conditions of this case are Figure 4 represents the numerical solutions of various WENO methods using 200 consistent cells with t = 2. We can see that the results of WENO-SIF are closer to the exact solution, while WENO-PZ shows nonphysical oscillations. That shows the present WENO-SIF is numerically stable and far less dissipative than the other WENO schemes.
In this example, we also compare the L 1 errors and CPU times of various schemes in computing this problem using different grid numbers, and the results are shown in Table 4 and Fig. 5. The results show the CPU times of WENO-SIF are significantly less than these WENO schemes when computing this problem. And all the errors of WENO-SIF are significantly smaller than those of the other schemes. That shows the WENO-SIF scheme has far better resolution and higher efficiency than others.

Case 2
The initial conditions of this case are where G(x, β, z) = e −β(x−z) 2 , F(x, α, a) = max(1 − α 2 (x − a) 2 , 0) and the constants z = −0.7, δ = 0.0005, β = ln2/(36δ 2 ) and α = 10. Figure 6 represents the numerical solutions of various WENO methods using 200 consistent cells with t = 6. In the square wave region (A), the results of both WENO-PZ and WENO-MN schemes show an upward jump, while the three schemes WENO-JS, WENO-Z, and WENO-M do not capture the square wave. In contrast, the present WENO-SIF is able to recognize the square wave and has a better match with the exact solution. And at the junction of square and triangular waves (B), the WENO-SIF provided in this paper performs significantly better than other WENO schemes. That shows the WENO-SIF scheme has better resolution than the others. We also compare the L 1 error and CPU time when computing this problem using different WENO schemes, and the results are presented in Table 5 and Fig. 7. That shows the CPU times of WENO-SIF are significantly less than the other WENO schemes, and the errors are also less than the other schemes. That shows the WENO-SIF scheme has good resolution and high efficiency than others.

One-dimensional inviscid Burgers equation
Next, we will solve the one-dimensional linear inviscid Burgers equation of the following form u t + uu x = 0, x ∈ [0, 2π], u 0 (x) = 0.5 + 0.5sin(x), periodic boundary. We also compare the L 1 error and CPU time for this problem, and the results are presented in Table 6 and Fig. 9. The results show that WENO-SIF requires significantly less CPU time than other WENO schemes to compute this problem with higher accuracy. That shows the WENO-SIF scheme has a higher resolution and computational efficiency than other WENO schemes.
Thereafter, we have also computed the one-dimensional advection equation and Burgers' equation after selecting several different values of λ, and the results are shown in Figs. 10, 11, and 12. Figure 10 depicts the results of case 1 with 200 cells at t = 2, the values of λ are 0.05, 0.10, 0.13, 0.20 and 0.50. One can see that numerical oscillations occur in the results when λ = 0.05 and 0.10, while the results for λ = 0.13 and 0.20 are significantly better than the rest. And we can see from Fig. 11, WENO-SIF will produce significant oscillations at the contact points of various waves when λ = 0.05, while the results for λ = 0.13 are better than the rest. In the results of the Burgers problem, oscillations occur for λ = 0.05, while the results for λ = 0.10 are better than the rest. Combining the above three arithmetic examples, it shows that the choice of λ = 0.13 for WENO-SIF is reasonable.

One-dimensional gas dynamic Euler equations
This section focuses on performing various WENO schemes for the 1-D gas dynamic Euler equation.

WENO-JS WENO-Z WENO-M WENO-PZ WENO-MN WENO-SIF
where ρ, u, p are the density, velocity and pressure, respectively. E = p/(γ − 1) + 1 2 ρu 2 is the total energy and γ is the specific heat ratio is set as γ = 1.4. And the time step is  In this example, we also compare the L 1 errors and CPU times of various schemes in computing this problem using different grid numbers, and the results are shown in Table 7 and Fig. 14. It is easy to see that the WENO-SIF scheme has the least CPU time, and its error is the smallest among the results of all WENO schemes. That shows the present WENO-SIF has optimal computational efficiency.

Shu-Osher's problem
The initial conditions of Shu-Osher's problem at [−5, 5] are as follows This problem contains low-frequency and high-frequency density disturbances and is used to test the performance of different WENO methods. Figure 15 presents the distribution curves of density for various WENO methods with 400 cells at t = 1.8. The reference solution is   Table 8 and Fig. 16. It is easy to see that the WENO-SIF scheme has the least CPU time, and its error is the smallest among the results of all WENO schemes. That shows the present WENO-SIF has optimal computational efficiency.

Titarev-Toro's problem
The initial conditions of the problem as follows   Figure 17 represents the computational results of various WENO methods using 2000 cells at t = 5. The reference solution is computed by using third-order WENO-JS at 8000 cells. See from the figure that WENO-SIF performs better than the other WENO schemes. The L 1 errors and CPU times for the problem on the selected grid are shown in Table 9 and Fig. 18. It is easy to see that the WENO-SIF scheme takes much less CPU time to compute the problem and its accuracy is higher than that of the others. That indicates the present WENO-SIF has the best resolution and computational efficiency.

Two dimension gas dynamic Euler equation
In this section, we will investigate the performance characteristics of the WENO scheme mentioned in this paper by solving a two-dimensional gas dynamics problem of the following form where, ρ, u, v, p are the density, x-velocity, y-velocity and pressure, respectively. And the total energy E is defined as And the time step is where c = √ γ p/ρ and C F L = 0.6. We will use the global Lax-Friedrichs flux splitting method in each of the following examples, and the numerical fluxes are reconstructed in the characteristic space. The specific heat ratio γ = 1.4 for all examples except γ = 5/3 in RT instability problem.

2-D Riemann problem: case 1
The initial conditions for two-dimensional Riemann problem are as following (Roe 1981 In the results, there is a mushroom-like structure symmetric about the diagonal y = x, and the richness of the vortex structure at the interface can usually examine the resolution of the different schemes. The boundary conditions for this problem are: all four edges are zero-order extrapolation. Figure 19 represents the density contours of this case by using 800 × 800 cells until t = 0.8 with 24 contours at [0.2, 1.7]. One can see that the unstable Kelvin-Helmholtz microstructures constructed by WENO-SIF are significantly more abundant than the other WENO schemes, showing that the present scheme has better resolution than the rest.
All four edges of the problem are zero-order extrapolation boundary conditions. Figure 20 shows the density contours for the various WENO schemes using 800 × 800 cells until
All four edges of the problem are reflection boundaries. Figure 21 shows the density contours for the various WENO schemes using 800 × 800 cells until t = 2.5 with 12 contours at [0.45, 1.0]. Comparing the jets on the diagonal in each graph, we can see that the length of WENO-SIF (F) is longer than others. That shows the present scheme is better than the rest in computing this problem.

Double-Mach reflection problem
This problem has widely used as a test example for high-order schemes. The initial conditions of double-Mach reflection problem on the computational region [0, 4] × [0, 1] given as Jiang and Shu (1996); Hong et al. (2020); Woodward and Colella (1984) The boundary conditions for this problem are: on the bottom, exact post-shock conditions are imposed in the interval [0, 1/6], and we use the reflection boundary for the rest. We set the top to be the exact motion of a Mach 10 shock. Inflow and outflow for the left and right, respectively. Figure 22 shows the density contours for the various WENO schemes using 1024 × 256 cells until t = 0.2 with 30 contours at [2,22]. At the interface, WENO-SIF has a richer vortex structure than the rest of the solutions. That shows the present WENO-SIF has a better resolution than others.
This problem serves as a benchmark to examine the performance of various schemes to capture complex vortex structures and preserve symmetry. We set the boundaries of the problem: the bottom and top are the incoming and outgoing flow boundaries, respectively, and the left and right sides are the reflection boundaries. Figure 23 shows the density contour profiles for various WENO methods using 256 × 1024 grids up to t = 1.95, 13 density contours at [0.9, 2.2]. Comparing the results of all the schemes, one can see that the present WENO-SIF has the richest vortex structure and preserves the symmetry.

Forward-facing step problem
The problem describes a Mach 3 supersonic fluid entering a tunnel with a step that is reflected several times on the surface, thereby generating a shock wave for the disembodied body. This problem is commonly used to test the numerical stability and resolution of different numerical schemes. The wind tunnel has a length of 3, a width of 1 and a height of 0.2 with a step of 0.6 in the wind tunnel. (ρ, u, v, p) = (1.0, 3.0, 0, 0, 0.71429) is on the left side of the wind tunnel, and outflow boundary conditions on the right side. The surfaces of both the tunnel and the step are reflection boundaries. Figure 24 shows the density contour profiles for various WENO methods using 900 × 300 grids at t = 4.0, 45 density contours at [0.5, 6.0]. At the interface of these shocks, the WENO-SIF has a richer vortex structure than the others. This indicates that the present WENO-SIF has a better resolution than other WENO schemes.
We set all bounds for this problem to be periodic. Figure 25 shows the density contours for the various WENO schemes using 800 × 800 cells until t = 1 with 12 contours at [0.9,2.2]. As can be seen in Fig., the vortex structure of the WENO-SIF scheme is larger than that of the remaining schemes. This indicates that the present scheme has a higher resolution than other schemes.

Shock/schear layer interaction problem
The initial conditions of the Shock/schear layer interaction problem at where b = 10, a 1 = a 2 = 0.05 and T = 30/2.68. The right boundary is the outer flow whereas the upper and lower boundaries are set as the post-shock and slip walls, respectively. Figure 26 represents the density contour profiles for various WENO methods using 600×120 It can be observed that the vortex structure of the WENO-SIF scheme is richer than that of the remaining schemes. This indicates that the present scheme has a higher resolution than the alternative schemes.

Computational time for 2-D problems
In this section, we present the time consumed by the various WENO methods for computing the 2D problem using a desktop computer with an Intel i7-9700 CPU @ 3.00 GHz processor, and we list the results in Table 10. In this table, we will take the time taken by WENO-JS to obtain the relative time of the various WENO schemes. Of all the WENO schemes, WENO-SIF uses the smallest amount of CPU time. Compared with the WENO-JS scheme, WENO-SIF saves at least about 20% of computation time. And the CPU time of WENO-SIF is even only 20% of WENO-PZ. Combining the individual one-and two-dimensional problems, the WENO-SIF scheme provided in this paper is computationally more efficient than other WENO schemes.

Conclusion
To improve the shock capturing and reduce the computational consumption of the third-order WENO-JS scheme, in this paper, we propose a new WENO scheme that is independent of the smoothing factor. The weight functions of the WENO-SIF are the segmented linear functions of the ratio to the substencil flux, which contain a tunable parameter λ. This function satisfies ω 0 (0) = 1 (ω 1 (0) = 0) and ω 0 (∞) = 0 (ω 1 (∞) = 1), which ensures the new present WENO scheme contains the ENO property. Numerical results show that the present WENO scheme can achieve the desired accuracy at high-order critical points, and the resolution of WENO-JS can be further improved by suppressing numerical oscillations near discontinuities at a low additional computational cost.