J Sci Comput DOI 10.1007/s10915-017-0587-4 TECHNICAL NOTE An Improved Third-Order WENO-Z Scheme Weizheng Xu1 · Weiguo Wu1 Received: 7 May 2017 / Revised: 19 September 2017 / Accepted: 19 October 2017 © Springer Science+Business Media, LLC 2017 Abstract In this paper, we develop an improved third-order WENO-Z scheme. Firstly, a new reference smoothness indicator is derived by slightly modifying that of WENO-N3 scheme proposed by Wu and Zhang (Int. J. Numer. Meth. Fl. 78:162–187, 2015). Then a new term is added to the weights of the developed scheme to further slightly increase the weight of less-smooth stencil. Some numerical experiments are provided to demonstrate that the improved scheme is stable and significantly outperforms the conventional third-order WENO scheme of Jiang and Shu, while providing essentially non-oscillatory solutions near strong discontinuities. Keywords WENO schemes · Smoothness indicators · High-resolution · Hyperbolic conservation laws Mathematics Subject Classification 65M12 · 65M70 · 41A10 1 Introduction The classical Weighted essentially non-oscillatory scheme (WENO-JS scheme) which is first proposed in [13] and then improved by Jiang and Shu [9] has become a quite successful methodology for solving problems with strong discontinuities and complicated smooth solutions in computational fluid dynamics. However, WENO-JS is too dissipative to solve problems with lots of structures in the smooth part of the solution, such as direct numerical simulation of turbulent flows. In [8], Henrick et al. find that this scheme fails to achieve the maximum convergence order of the scheme at critical points where the first derivative van- B Weizheng Xu [email protected] Weiguo Wu [email protected] 1 Key Laboratory of High Performance Ship Technology, Wuhan University of Technology, Wuhan 430063, China 123 J Sci Comput ishes but the third order derivative does not. In order to amend this drawback, an improved fifth-order WENO scheme (called WENO-M) has been suggested by incorporating a mapping procedure to maintain the nonlinear weights of the convex combination of stencils as near as possible to the linear weights of fifth order accuracy except for discontinuities. Inspired by the study in [8], Borges et al. [2] designed another fifth-order WENO scheme (called WENO-Z) by adding a higher order reference new smoothness indicator which is obtained by a linear combination of the classical smoothness indicator of WENO-JS. The new scheme can achieve superior results with almost the same computational effort of the classical WENO method. Castro et al. [3] developed a general formula for the higher order smoothness indicators and extended the WENO-Z scheme to all odd orders of accuracy. Several approaches have been considered for the improvement of the WENO-Z scheme in recent years. One of the effective approaches is through the design of the smoothness indicators [4–7,19–21]. Aforementioned articles mainly focus on improving the order of accuracy of the WENO-Z scheme at critical points. However, It should be mentioned that the common characteristic of improved schemes (WENO-Z and WENO-M schemes) is that they both assign larger weights to less-smooth stencils and while keep the ENO property. And such a strategy indeed decreases the dissipation around the discontinuous region. Acker et al. [1] present an improved fifth-order WENO-Z scheme (called WENO-Z+ scheme) through adding a new term to the fifth-order WENO-Z weights to further increase the relevance of less-smooth substencils. The improved scheme attains much better resolution at the smooth parts of the solution, while keeping the same numerical stability of the original WENO-Z at shocks and discontinuities. Based on the idea that increasing the contribution of the less smooth substencils can improve the performance of the conventional WENO-Z scheme, in the present work, we firstly devised a new reference smoothness indicator for the third-order WENO-Z scheme through slightly modifying that of the WENO-N3 scheme proposed in [20]. In fact, the purpose of this modification is to slightly increase the weight of less-smooth stencil. Then a new term was added to the weights of the developed scheme to further improve its performance. The outline of this paper is given as follows. We present, in Sect. 2, a brief description of WENO reconstructions and the conventional WENO-JS3 and WENO-Z3 scheme. Section 3 introduces the constructions of the improved third-order WENO-Z scheme. Some numerical examples are provided in Sect. 4 to demonstrate advantages of the proposed WENO-Z scheme. Finally, concluding remarks are given in Sect. 5. 2 WENO Schemes The one-dimensional hyperbolic conservation law in the form ∂u ∂ f (u) + = 0, (1) ∂t ∂x can be approximated by the semi-discrete form: du i ∂ f (u) =− (2) x=xi , dt ∂x The conservation property of the spatial discretization is obtained by defining an implicit function H (x) through the following equation 1 x+h/2 f (u) = H (ξ )dξ, (3) h x−h/2 123 J Sci Comput where h denotes the uniform grid size. The spatial derivative in Eq. (2) is exactly represented as a conservative finite difference formula at the cell boundaries. A conservative finite difference formulation looks for numerical flux approximating H (xi±1/2 ) at the cell boundaries with higher accuracy. In practice, the approximation to the spatial derivative ∂ f /∂ x at xi will take the conservative form f i+1/2 − f i−1/2 ∂ f (u) , x=xi ≈ ∂x h (4) Then the following relation can be obtained f i+1/2 − f i−1/2 du i = L (u i ) = − . dt h (5) 2.1 WENO-JS3 Scheme The third-order WENO scheme uses two candidate stencils. Thus the numerical flux is approximated by a convex combination of the second-order fluxes over each stencil, f i+1/2 = ω0 f 0,i+1/2 + ω1 f 1,i+1/2 , (6) 1 3 1 1 f 0,i+1/2 = − f i−1 + f i , f 1,i+1/2 = f i + f i+1 . 2 2 2 2 (7) where To achieve the high order accuracy in the smooth regions, the WENO approximation should simulate the central upwind scheme of optimal order, while retaining the ENO property near discontinuities. Therefore, the WENO methods design the weight functions, ωi , to be nonlinear in order to change dynamically based on the smoothness of the numerical solutions. The classical weight functions proposed by Jiang and Shu [9] are as follows: ωk = αk dk , αk = , k = 0, 1, 1 (ε + βk )2 αs (8) s=0 where the parameter ε in Eq. (8) is the small positive number that is used to avoid the division by zero. It is set to be 10−6 in the present work, as recommended in [9]. dk is the optimal weight: d0 = 1 2 , d1 = , 3 3 (9) and the βk is the smoothness indicator: β0 = ( f i−1 − f i )2 , β1 = ( f i − f i+1 )2 . (10) 2.2 WENO-Z3 Scheme For the third-order WENO-Z scheme (WENO-Z3), Don and Borges [5] considered the reference smoothness indicator of the form τ , k = 0, 1, (11) αk = dk 1 + βk + ε τ Z = |β0 − β1 | , (12) 123 J Sci Comput The authors in [20] have previously conducted some derivations on the accuracy of the WENO-Z3 scheme at the non-critical points. However, they use the following Eq. (13) to calculate the nonlinear weights ωk . dk 1 + βτk τ αk = d 1 + (13) = ωk = k 1 1 βk τ αs ds 1 + βs s=0 s=0 In fact, the Eq. (13) is equal to functions αk not the nonlinear weights ωk . Then, they concluded that the WENO-Z3 scheme can not achieve the convergence order at the non-critical points according to the sufficient condition as stated in [3,5,6,20,21]. ωk − d k = o h 2 . (14) Now, we will again present a short derivation to investigation on the accuracy of the WENOZ3 scheme at the non-critical points in the smooth flow fields with the way of Taylor expansion. The smoothness indicators of the candidate stencils in Eq. (10) can be expanded as a Taylor series at xi ⎧ ⎨ β0 = f 2 h 2 − f f h 3 + 1 f 2 + 1 f f h 4 − fi fi h 5 − fi fi h 5 + o h 6 , i i i 4 i 3 i i 12 6 (15) 1 2 1 4 f i f i 5 f i f i 5 ⎩ 2 2 3 β1 = f i h + f i f i h + 4 f i + 3 f i f i h + 12 h + 6 h + o h 6 , At the non-critical points, substituting Eqs. (15) and (12) into Eq. (11) gives 3 2 f i2 2 2 f i τZ = d0 1 + h + 2 h + o h α0 = d0 1 + β0 fi fi 3 2 f i2 2 2 f i τZ = d1 1 + h − 2 h + o h α1 = d1 1 + β1 fi fi (16) (17) Then, with the weighting procedure in Eq. (8), we can obtain the following relation ω0 = α0 1 8 f i2 2 = + h + o h3 α0 + α1 3 9 f i2 (18) ω1 = α1 2 8 f i2 2 = − h + o h3 2 α0 + α1 3 9 fi (19) Thus the numerical flux f i+1/2 in Eq. (5) is given by 1 1 2 1 3 1 (4) 4 f i3 f i+1/2 = f i + f i h + h4 + o h5 fi h + fi h + fi − 2 2 12 12 144 9 fi Similarly, the numerical flux f i−1/2 is given by f i−1/2 1 1 2 1 3 = f i − f i h + f h + f h − 2 12 i 12 i 11 (4) 4 f i4 f + 144 i 9 f i2 h4 + o h5 (20) (21) Finally we get the following results f i+1/2 − f i−1/2 h 123 = f i + 1 (4) 3 f i h + o h 4 = f i + o h 3 12 (22) J Sci Comput From above results, it is clear that the WENO-Z3 scheme can achieve 3rd-order accuracy at the smooth flow field excluding the critical points. 2.3 Time Discretization After the spatial derivative is discretized with one of the WENO schemes, an ordinary differential equation (ODE) system is obtained as follow du = L (u) , dt (23) where the spatial operator L(u) is represented in Eq. (5). In all the numerical simulations in this paper, Shu’s 3rd order TVD time stepping method [9] is applied for the unsteady time integrations of the ODE system. The third-order TVD Runge–Kutta method is written as: ⎧ n n (1) ⎪ ⎨ u = u + t L (u ) , 3 1 (24) u (2) = 4 u n + 4 u (1) + 41 t L u (1) , ⎪ ⎩ n+1 1 n 1 (2) 2 (2) = 3 u + 3 u + 3 t L u . u 3 Improved WENO Scheme 3.1 WENO-P3 Scheme In reference [20], Wu and Zhao proposed the WENO-N3 scheme by introducing the new reference smoothness indicator of the form β0 + β1 τ N = (25) − β3 , 2 where β3 is the smoothness indicator of the whole stencil {xi−1 , xi , xi+1 } which is expressed as [6] β3 = 13 1 ( f i−1 − 2 f i + f i+1 )2 + ( f i−1 − f i+1 )2 . 12 4 In the present work, a new smoothness indicator is derived in such a way β0 + β1 1 2 τP = − ( f i−1 − f i+1 ) , 2 4 (26) (27) For simplicity, we denote the WENO scheme with the new smoothness indicator as WENOP3 scheme. 3.1.1 Local Accuracy of the WENO-P3 Scheme In order to investigate on the characteristics of the WENO-P3 scheme, firstly, we give a short derivation to investigate on the accuracy of the improved WENO-P3 scheme at the critical points. 123 J Sci Comput At the first-order critical point ( f i = 0, f i = 0, f i = 0), substituting Eqs. (15) and (27) into Eq. (11) gives 2 τp 2 f i α0 = d0 1 + = d0 2 + h + o h (28) β0 3 f i 2 τp 2 f i = d1 2 − h + o h (29) α1 = d1 1 + β1 3 f i Then, with the weighting procedure in Eq. (8), we can obtain the following relation ω0 = α0 1 α1 2 4 f i 4 f i = + h + o h 2 , ω1 = = − h + o h2 α0 + α1 3 27 f i α0 + α1 3 27 f i (30) Thus the numerical flux f i+1/2 in Eq. (5) is given by 1 1 1 2 2 3 1 (4) 4 f i+1/2 = f i + f i h + fi h + fi − fi h + fi h + o h 5 2 12 12 27 144 (31) Similarly, the numerical flux f i−1/2 is given by f i−1/2 = f i − 1 13 2 fi h + fi h + o h 3 2 92 (32) Finally we get the following results f i+1/2 − f i−1/2 h = f i − 55 f h + o h 2 = f i + o (h) 276 i (33) Thus, the WENO-P3 methods only achieve 1rd-order accuracy at critical points. 3.1.2 Weights of Less-Smooth Substencils of the WENO-P3 Scheme In this subsection, we give the comparison of the weights of less-smooth substencils between the WENO-P3 scheme and WENO-N3 scheme. After some symbolic derivations, the following relation can be obtained β0 + β1 10 τ N = (34) − β3 = ( f i − 2 f i−1 + f i+1 )2 2 12 β0 + β1 3 1 − ( f i−1 − f i+1 )2 = (35) τ P = ( f i − 2 f i−1 + f i+1 )2 2 4 12 Proposition Suppose SC and S D are two substencils of the same stencil, such the that ωD solution is smoother at SC than at S D (meaning that βC < β D ). Then ωC > τP ωD , (τ N > τ P ). ωC τN Proof Firstly we have the following relation disregarding ε ωD αD αD ωD dD 1 + − = − = ωC τ P ωC τ N αC τ P αC τ N dC 1 + 123 τP βD τP βC − 1+ 1+ τN βD τN βC (36) J Sci Comput Since β D > βC , we have a = β D , b = βC ⇒ 0 < b < a Therefore 1+ 1+ τP a τP b − 1+ 1+ τN a τN b = (a − b) (τ N − τ P ) >0 ab 1 + τbP 1 + τbN (37) From Eqs. (36) and (37), we can obtain the conclusion ωD ωC τP ωD > ωC τN , (τ N > τ P ) (38) From above conclusion, it is obvious that WENO-P3 scheme is expected to be less dissipative giving better resolution results than the WENO-N3 scheme. And, this conclusion is in agreement with the following simulation results. 3.2 WENO-P+3 Scheme It is demonstrated in [1] that, at relatively coarse grids, increasing the weights of less-smooth stencils is the most relevant cause than the order of accuracy at critical points for better resolution of waves. Acker et al. proposed the fifth-order WENO-Z+ scheme based upon this idea, which shows substantial improvement on the numerical resolution of problems containing shocks and high gradients. Here, we apply this new technique to WENO-P3 scheme to further increase the weights of less smooth substentils, whereas, the function αk is defined by βk + ε τP +λ , k = 0, 1, (39) αk = dk 1 + βk + ε τP + ε β0 + β1 1 − ( f i−1 − f i+1 )2 , ε = 10−40 . (40) τ P = 2 4 where λ is a parameter for fine-tuning the size of the increment of the weight of less smooth stencils. The value of λ was empirically determined to be λ = h 1/6 as the one yielding the best results in terms of stability and resolution power for the standard tests suite, as we shall see in Sects. 4.1.5, 4.2.3, 4.2.4, and 4.2.5. 3.2.1 Local Accuracy of the WENO-P+3 Scheme Firstly, we give a short derivation to investigate on the accuracy of the improved WENO-P+3 scheme at the critical points. At the first-order critical point ( f i = 0, f i = 0, f i = 0), substituting Eqs. (15) and (40) into Eq. (39) gives τp 2 f i β0 α0 = d0 1 + = d0 2 + λ + +λ h + o (41) (λh) β0 τp 3 f i τp β1 2 f i +λ h + o = d1 2 + λ − α1 = d1 1 + (42) (λh) β1 τp 3 f i 123 J Sci Comput Then, with the weighting procedure in Eq. (8), we can obtain the following relation ω0 = α0 1 α1 2 4 f i 4 f i = + h + o , ω = = h + o (λh) − (λh) 1 α0 + α1 3 27 f i α0 + α1 3 27 f i (43) Finally we get the following results f i+1/2 − f i−1/2 h = f i − 55 f i h + o h 2 = f i + o (h) 276 (44) Thus, the WENO-P+3 methods only achieve 1rd-order accuracy at critical points. From Eqs. (33) to (44), we can see that the WENO-P+3 almost share the same accuracy with the WENO-P3 scheme. It is also indicated that the parameter λ almost has no influence on the accuracy of the WENO-P+3 scheme. 3.2.2 Weights of Less-Smooth Substencils of the WENO-P3 Scheme Similarly, we still give a short derivation to verify that the WENO-P+3 scheme puts a relatively larger weight on less-smooth substencils than WENO-P3 scheme. Proposition Suppose SC and S D are two substencils of the same stencil, such that the solution is smoother at SC than at S D (meaning that βC < β D ). Then ωωCD > W E N O−P+3 ωD . ωC W E N O−P3 Proof Firstly we have the following relation disregarding the parameter ωD ωD − ωC W E N O−P+3 ωC W E N O−P3 βD τP 1+ αD αD dD 1 + βD + λ τP = − = − β τ C P αC W E N O−P+3 αC W E N O−P3 dC 1 + 1+ βC + λ τ P τP βD τP βC (45) Since β D > βC , we have A= τP τP ,B = ⇒B> A>0 βD βC Therefore 1+ A+ 1+ B+ λ A λ B − 1+ A (B + A + 1) >0 = λ (B − A) 1+ B AB 1 + B + Bλ (1 + B) Considering Eqs. (45) and (46), we can obtain the conclusion ωD ωD > ωC W E N O−P+3 ωC W E N O−P3 (46) (47) From above conclusion, it can be expected that WENO-P+3 scheme will give better resolution results than the WENO-P3 scheme. And, this conclusion is in agreement with the following simulation results as well. 123 J Sci Comput 4 Numerical Results In this section, we consider several benchmark problems to illustrate the improvement produced by the present method. 4.1 Scalar Test Problems In order to observe the shock capturing ability of the proposed WENO-P and WENO-P+3 schemes, we test them for one dimensional scalar advection equations with five initial data. The results of the proposed scheme are compared to those of WENO-JS3, WENO-N3 and the classical WENO-JS5 scheme [9]. Let us first consider the one-dimensional linear advection equation controlled by ∂u ∂t + ∂∂ux = 0, −1 < x < 1 u (x, 0) = u 0 (x) , periodic (48) 4.1.1 Critical Point Problem 1 In this case, the initial condition in [8] is chosen as follow sin (π x) u 0 (x) = sin π x − π (49) which contains two first-order critical points at which f = 0 and f = 0. The L 1 and L ∞ errors along with the numerical order of accuracy are provided in Tables 1 and 2 for WENO-JS3, WENO-Z3, WENO-N3 and WENO-P3 schemes. 4.1.2 Critical Point Problem 2 In this case, the initial condition [6] is as follow u 0 (x) = sin3 (π x) Table 1 A comparison study of L 1 (error and order) for linear advection equation with initial condition (49) at t = 2 N WENO-JS3 (50) WENO-Z3 L 1 (error) L 1 (order) L 1 (error) L 1 (order) 25 1.2429E−1 – 5.7276E−2 – 50 4.6050E−2 1.4324 1.5116E−2 1.9219 100 1.3010E−2 1.8236 3.4645E−3 2.1254 200 1.5132E−3 3.1039 8.3721E−4 2.0490 400 1.4618E−4 3.3718 1.6823E−4 2.3152 N WENO-N3 WENO-P3 L 1 (error) L 1 (order) L 1 (error) L 1 (order) 25 4.8254E−2 – 3.2863E−2 – 50 1.1315E−2 2.0924 6.7032E−3 2.2935 100 2.5549E−3 2.1469 1.3325E−3 2.3307 200 4.9809E−4 2.3588 2.2705E−4 2.5531 400 1.0067E−4 2.3068 4.1760E−5 2.4428 123 J Sci Comput Table 2 A comparison study of L ∞ (error and order) for linear advection equation with initial condition (49) at t = 2 N 25 WENO-JS3 L ∞ (order) L ∞ (error) L ∞ (order) 2.6012E−1 – 1.4836E−1 - 50 1.1014E−1 1.2398 5.6752E−2 1.3864 100 4.6113E−2 1.2561 2.0758E−2 1.4510 200 1.0219E−2 2.1739 7.4819E−3 1.4722 400 1.6899E−3 2.5962 2.5623E−3 1.5460 N WENO-N3 L ∞ (error) Table 3 A comparison study of L 1 (error and order) for linear advection equation with initial condition (50) at t = 2 WENO-Z3 L ∞ (error) WENO-P3 L ∞ (order) L ∞ (error) L ∞ (order) 25 1.2411E−1 – 9.7985E−2 – 50 4.7462E−2 1.3868 3.1544E−2 1.6352 100 1.6716E−2 1.5055 9.5830E−3 1.7188 200 5.7271E−3 1.5454 2.7884E−3 1.7810 400 1.9697E−3 1.5398 8.4490E−4 1.7226 N WENO-JS3 WENO-Z3 L 1 (error) L 1 (order) L 1 (error) L 1 (order) 25 1.8499E−1 – 1.5221E−1 – 50 9.1142E−2 1.0213 4.1257E−2 1.8834 100 2.9843E−2 1.6107 9.1524E−3 2.1724 200 7.3790E−3 2.0159 2.0387E−3 2.1665 400 7.0804E−4 3.3815 4.1500E−4 2.2965 N WENO-N3 WENO-P3 L 1 (error) L 1 (order) L 1 (error) L 1 (order) 25 1.4067E−1 – 1.2275E−1 – 50 3.2229E−2 2.1259 2.3281E−2 2.3985 100 6.7468E−3 2.2561 4.2600E−3 2.4502 200 1.4009E−3 2.2678 8.0490E−4 2.4040 400 2.7477E−4 2.3501 1.4483E−4 2.4744 which contains a first-order critical point at which f = 0, f = 0, but f = 0. Tables 3 and 4 contain the L 1 and L ∞ errors along with the numerical order of accuracy for WENOJS3, WENO-Z3, WENO-N3 and WENO-P3 schemes. From Tables 1, 2, 3, and 4, it is clear that the WENO-P3 scheme almost share the same accuracy with the WENO-Z3 and WENON3 scheme, which is in agreement with the discussions in Sects. 3.1.1 and 3.2.1. 4.1.3 Linear Discontinuity Problem A solution with an initial discontinuity [2] is tested − sin (π x) − 21 x 3 , u 0 (x) = − sin (π x) − 21 x 3 + 1, 123 −1 < x ≤ 0 0<x ≤1 (51) J Sci Comput Table 4 A comparison study of L ∞ (error and order) for linear advection equation with initial condition (50) at t = 2 N 25 WENO-JS3 WENO-Z3 L ∞ (error) L ∞ (order) L ∞ (error) L ∞ (order) 3.8696E−1 – 3.1669E−1 – 50 2.2025E−1 0.8130 1.2435E−1 1.3487 100 9.7458E−2 1.1763 4.8963E−2 1.3446 200 3.6946E−2 1.3994 1.7450E−2 1.4885 400 7.1187E−3 2.3757 6.1192E−3 1.5118 N WENO-N3 L ∞ (error) WENO-P3 L ∞ (order) L ∞ (error) L ∞ (order) 25 2.9180E−1 – 2.5016E−1 – 50 1.0164E−1 1.5215 7.1777E−2 1.8013 100 4.0100E−2 1.3418 2.8144E−2 1.3507 200 1.3554E−2 1.5649 8.5412E−3 1.7203 400 4.5504E−3 1.5747 2.6072E−3 1.7119 1.2 Exact WENO-JS3 WENO-N3 WENO-P3 WENO-P+3 WENO-JS5 1.0 0.8 u 0.6 0.4 0.2 0.0 -0.2 -1.0 -0.5 0.0 0.5 1.0 x Fig. 1 Numerical results with the initial condition (51), t = 6 which is a piecewise sine function with a jump discontinuity at x = 0. We solve it up to t = 6 to see the behaviors of the WENO schemes at the jump discontinuity. Numerical solutions for the WENO-JS3, WENO-N3, WENO-P3, WENO-P+3 and WENO-JS5 schemes with grid number N = 200 at the final time are shown in Fig. 1. It is clear that, near the discontinuity, the WENO-P+3 scheme obtains more sharper solution than the WENO-JS3, WENO-N3 schemes. WENO-P3 scheme is slightly better than WENO-N3 scheme due to slightly larger weight assignment to the discontinuous stencils as shown in Fig. 2b and c. What is more, WENO-P+3 does not perform better than the WENO-JS5 due to the accuracy loss at the critical points as demonstrated in Sect. 3.1.1. 123 J Sci Comput 0 0 10 10 -1 -1 10 10 -2 -2 10 10 -3 -3 10 10 -4 -4 10 10 ω0 ω1 -5 10 ω0 ω1 -5 10 d0 d0 d1 -6 10 -7 10 d1 -6 10 -7 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 10 -0.04 -0.03 -0.02 -0.01 x (a) 0.02 0.03 0.04 (b) 0 10 10 -1 -1 10 -2 10 -3 10 10 -2 10 -3 10 -4 -4 10 10 ω0 ω1 -5 10 ω0 ω1 -5 10 d0 -6 d0 -6 d1 10 d1 10 -7 -7 -0.04 0.01 x 0 10 0.00 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 10 -0.04 -0.03 -0.02 -0.01 0.00 x x (c) (d) 0.01 0.02 0.03 0.04 Fig. 2 a WENO-JS3. b WENO-N3. c WENO-P3. d WENO-P+3. The distribution of the ideal weights dk and the weights ωk for the schemes at the first step of the numerical solution of the nonlinear discontinuity problem Figure 2 shows the weights ωk for the WENO-JS3, WENO-N3, WENO-P3 and WENOP+3 schemes at the first step of the numerical solution of the wave equation with the initial condition (51). The ideal weights dk are also plotted as lines and the vertical axis is in log10 scale. And from this, one can observe that WENO-N3, WENO-P3 and WENO-P+3 assign larger weights for the discontinuous stencils as compared with WENO-JS3. The WENO-P3 scheme assigns larger weights for the discontinuous stencils as compared with WENO-N3 scheme. The WENO-P+3 scheme further assigns larger weights for the discontinuous stencils as compared with WENO-P3 due to be added term. From above analysis, it can be concluded that further increasing the relevance of less-smooth stencils can give less dissipative simulation results around discontinuities and smooth flows as demonstrated in [1]. 4.1.4 Complex Wave A more complex initial solution is tested in [14] given by ⎧ −1 ≤ x < −1/3 ⎨ −x sin 3π x 2 /2 , u 0 (x) = |sin (2π x)| , −1/3 ≤ x < 1/3 ⎩ 2x − 1 − sin (3π x)/6, other wise 123 (52) J Sci Comput 1.0 u 0.5 0.0 Exact WENO-JS3 WENO-N3 WENO-P3 WENO-P+3 WENO-JS5 -0.5 -1.0 -1.0 -0.5 0.0 0.5 1.0 x Fig. 3 Numerical results with the initial condition (52), t = 6 The solution of this problem contains contact discontinuities, corner singularities and smooth areas. We solve the equation up to t = 6 with grid number N = 200. The numerical results are provided in Fig. 3. Comparison of the results demonstrates the advantages of the proposed WENO-P3 and WENO-P+3 schemes to approximate the exact solution near discontinuities as well as critical points. However, the WENO-P+3 shows worse performances compared with the WENO-JS5 at the critical points and the sharp points due to the accuracy loss. 4.1.5 Gaussian-Square-Triangle-Ellipse Linear Test Then, a more challenging test case [12] that contains a smooth combination of Gaussians, a square wave, a sharp triangle wave, and a half ellipse is calculated. ⎧1 ⎪ 6 (G (x, β, z − δ) + G (x, β, z + δ) + 4G (x, β, z)) , − 0.8 ≤ x ≤ − 0.6 ⎪ ⎪ ⎪ − 0.4 ≤ x ≤ − 0.2 ⎨ 1, 0 ≤ x ≤ 0.2 (53) u 0 (x) = 1 − |10 (x − 0.1)| , ⎪ 1 ⎪ ⎪ (F (x, α, a − δ) + F (x, α, a + δ) + 4F (x, α, a)) , − 0.8 ≤ x ≤ − 0.6 ⎪ 6 ⎩ 0, other wise where G (x, β, z) = e−β(x−z) , F (x, α, a) = 2 max 1 − α 2 x − a 2 , 0 (54) The constants are taken as a = 0.5, z = − 0.7, d = 0.005, α = 10, β = 1og2/(36δ 2 ). We solve the problem on a 200 grid points uniform mesh and the results of WENO-P3 and WENO-P+3 schemes are compared against the WENO-JS3, WENO-N3 and WENO-JS5 schemes until the final time t = 8 as shown in Fig. 4. It can be observed that the WENO-P+3 scheme achieves better resolution of waves than the WENO-JS3, WENO-N3 schemes. The WENO-P+3 gives poorer results than the WENO-JS5 particularly at the sharp points due to the accuracy loss. Figure 5 compares the results of this test with N = 200 points for the WENO-P+3 scheme with five different values for the parameter λ: h 1/6 , h 1/4 , h 1/2 , h 2/3 , and h 3/3 . From Fig. 5, 123 J Sci Comput Exact WENO-JS3 WENO-N3 WENO-P3 WENO-P+3 WENO-JS5 1.0 0.8 u 0.6 0.4 0.2 0.0 -1.0 -0.5 0.0 0.5 1.0 x Fig. 4 Numerical solution of the advection equation with the discontinuous initial condition (53) Exact λ = h0 λ = h1/6 λ = h1/4 λ = h1/2 λ = h2/3 λ = h3/3 1.0 0.8 u 0.6 0.4 0.2 0.0 -1.0 -0.5 0.0 0.5 1.0 x Fig. 5 Numerical solution of the advection equation with the discontinuous initial condition (53) using the WENO-P+3 scheme with different values of λ we can see that the WENO-P+3 scheme with parameter λ = h 1/6 performs the best. It should be mentioned that the WENO-P+3 scheme will become unstable with more larger λ. That is to say that, the added term should be kept small enough for avoiding spurious oscillations and instabilities. From above simulation results, it is clear that WENO-P+3 scheme is less dissipative with larger value of the parameter λ, which indicates that WENO-P+3 scheme generally places a larger weight to less-smooth substencils with larger value of λ. Let us now verify this inference. 123 J Sci Comput Proposition Suppose SC and S D are two substencils of the same stencil, such that the solution is smoother at SC than at S D (meaning that βC < β D ). Then λ ω D1 λ ωC1 > λ ω D2 λ ωC2 , (λ1 > λ2 ). Proof Firstly we have the following relation disregarding ε ωλD1 ωλD2 Since β D > = α λD1 − α λD2 = ωCλ2 αCλ1 αCλ2 ⎫ ⎧ βD τ 1 + βτD + λ2 βτD ⎬ d D ⎨ 1 + β D + λ1 τ − = dC ⎩ 1 + τ + λ βC 1 + βτC + λ2 βτC ⎭ 1 βC τ ωCλ1 − (55) βC , we have a= τ τ ,b = ⇒0<a<b βD βC Therefore 1+a+ 1+b+ λ1 a λ1 b − 1+a+ 1+b+ λ2 a λ2 b = (b − a) (b + a + 1) (λ1 − λ2 ) >0 ab 1 + b + λ1 1 + b + λ2 b (56) b From Eqs. (55) to (56), we can obtain the conclusion ωλD1 ωCλ1 > ωλD2 ωCλ2 , (λ1 > λ2 ) (57) 4.2 One-Dimensional Euler Systems In this subsection, we present numerical experiments with the one dimensional system of the Euler equations for gas dynamics in strong conservation form: ∂U ∂F + =0 ∂t ∂x with (58) ⎤ ⎤ ⎡ ρu ρ 1 U = ⎣ ρu ⎦ , F = ⎣ ρu 2 + p ⎦ , p = (γ − 1) E − ρu 2 , γ = 1.4 2 E u (E + p) ⎡ And ρ, u, p, and E are the density, velocity, pressure and total energy, respectively. γ is the ratio of specific heats. The eigenvalues of Jacobian matric A (U ) = ∂ F/∂U can be written as follows: √ λ1 = u − c, λ2 = u, λ3 = u + c where c = γ p/ρ is the sound speed. The characteristic decomposition with Roe’s approximation is used at the cell faces and the Lax–Friedrichs formulation is used for the numerical fluxes to generalize the WENO schemes to the one dimensional Euler systems. 4.2.1 Sod Problem The initial data for the Sod problem [16] are given by 123 J Sci Comput Exact WENO-JS3 WENO-N3 WENO-P3 WENO-JS5 WENO-P+3 1.0 Density 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 x (a) 0.45 Exact WENO-JS3 WENO-N3 WENO-P3 WENO-JS5 WENO-P+3 Density 0.40 0.35 0.30 0.25 0.62 0.64 0.66 0.68 0.70 0.72 x (b) Fig. 6 Sod problem: a density distribution and b the enlarged portion near the discontinuity (ρ, u, p) = (1, 0, 1), (0.125, 0, 0.1), 0 ≤ x < 0.5 0.5 ≤ x ≤ 1 (59) We compute the equation until t = 0.18 with N = 200 points. The simulated density and enlarged portion near the discontinuity are shown in Fig. 6. The ’exact’ solution is the result computed on N = 10,000 points using the WENO-JS3 scheme. The performance of WENO-P3 yields better results than WENO-N3 and WENO-JS3 scheme. The WENO-P+3 scheme gives the least dissipative results even better than the WENO-JS5 scheme. 123 J Sci Comput 4.2.2 Lax Problem For the Lax problem [10], the initial conditions are given by (0.445, 0.698, 3.528), − 5 ≤ x < 0 (ρ, u, p) = (0.5, 0, 0.571), 0≤x ≤5 (60) and the final time is t = 1.3. The WENO-P+3 scheme is less dissipative than the other schemes and provides better resolution of the contact discontinuities and the shock waves, while maintaining the ENO property as shown in Fig. 7. 4.2.3 Shu–Osher Problem For the 1D Shu–Osher problem, the initial condition is (3.857143, 2.629369, 10.33333) , − 5 ≤ x < −4 (ρ, u, p) = −4 ≤ x ≤ 5 (1 + 0.2 sin 5x, 0, 1) , (61) We solve this problem up to time t = 1.8. This problem describes the phenomena that a shock-wave (Mach 3) propagating to the right interacts with a perturbed density disturbance. Figure 8 provides a comparison for all schemes with N = 600 points. The reference solution has been computed on a fine grid of 10,000 points using the WENO-JS3 scheme. It can be seen that the WENO-P+3 scheme captures the fine scale structures, especially at the high-frequency waves behind the shock, better than the WENO-JS3 and WENO-N3 scheme. However, WENO-JS5 scheme shows more accuracy results than the WENO-P+3 scheme particularly at the high-frequency waves. This is mainly due to the accuracy loss of the WENO-P+3 scheme at the critical points. Figure 9 compares the results of the Shu–Osher test with N = 600 points for the WENOP+3 scheme with five different values for the parameter λ: h 1/6 , h 1/4 , h 1/2 , h 2/3 , and h 3/3 . From Fig. 9, we can see that the choice λ = h 1/6 gives less dissipative results while the choice λ = h 3/3 gives more dissipative results. 4.2.4 Titarev–Toro Problem This is a variation of the Shu–Osher test, with a different initial condition (1.515695, 0.523346, 1.805000) , −5 ≤ x < −4.5 (ρ, u, p) = −4.5 ≤ x ≤ 5 (1 + 0.1 sin (20π x) , 0, 1) , (62) which consists of a right-facing shock wave of Mach number 1.1 running into a highfrequency density perturbation. The flow contains physical oscillations which have to be resolved by the numerical method. We compute the solution at the output time t = 5. Figure 10 provides graphical results for all schemes on a mesh of 3000 cells. The reference solution is obtained by applying the WENO-P+3 scheme on a fine mesh of 8000 cells and is shown by the solid line on all figures. It can be seen from Fig. 10 that the WENO-P+3 scheme resolves most of the waves with a good approximation to their amplitudes better than the WENO-JS3 and WENO-N3 schemes. Again, WENO-JS5 scheme shows more accuracy results than the WENO-P+3 scheme particularly at the high-frequency waves. Figure 11 compares the results of the Titarev–Toro test with N = 3000 cells for the WENO-P+3 scheme with the five different values for the parameter λ: h 1/6 ,h 1/4 h 1/2 , h 2/3 , and h 3/3 . We see that the choice λ = h 1/6 gives the best results, resolving most of the waves with a good approximation to their amplitudes. 123 J Sci Comput 1.4 1.2 Exact WENO-JS3 WENO-N3 WENO-P3 WENO-JS5 WENO-P+3 Density 1.0 0.8 0.6 0.4 0.2 -5 -4 -3 -2 -1 0 1 2 3 4 5 x (a) 1.4 1.2 Density 1.0 Exact WENO-JS3 WENO-N3 WENO-P3 WENO-JS5 WENO-P+3 0.8 0.6 0.4 1.2 1.6 2.0 2.4 2.8 3.2 3.6 x (b) Fig. 7 Lax problem: a density distribution and b the enlarged portion near the discontinuity 4.2.5 Interacting Blast Wave The one dimensional blast waves interaction problem of Woodward and Collela [18] has the following initial condition, with reflective boundary conditions on both ends ⎧ ⎨ (1, 0, 1000) , (ρ, u, p) = (1, 0, 0.01) , ⎩ (1, 0, 100) , 123 0 ≤ x < 0.1 0.1 ≤ x ≤ 0.9 0.9 ≤ x ≤ 1 (63) J Sci Comput 5.0 4.5 4.0 3.5 Density 3.0 2.5 Exact WENO-JS3 WENO-N3 WENO-P3 WENO-P+3 WENO-JS5 2.0 1.5 1.0 0.5 -5 -4 -3 -2 -1 0 1 2 3 4 5 x (a) 4.8 4.4 4.0 Density 3.6 3.2 Exact WENO-JS3 WENO-N3 WENO-P3 WENO-P+3 WENO-JS5 2.8 2.4 2.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 x (b) Fig. 8 Shu–Osher problem: a density distribution and b the enlarged portion of the high frequency waves region The initial pressure gradients generate two density shock waves that collide and interact later in time, forming a profile as shown in Fig. 12 on uniform grid of 600 at t = 0.038. The reference solutions are the numerical solutions of the WENO-JS3 scheme with grid points of N = 10,000. Careful examination of Fig. 12, it can be concluded that WENO-P+3 solution are much better resolved as compared with other schemes counterpart. Figure 13 compares the results of the Interacting blast wave problem with N = 600 points for the WENO-P+3 scheme with the five different values for the parameter λ: h 1/6 ,h 1/4 h 1/2 , h 2/3 , and h 3/3 . It is clear that the choice λ = h 1/6 gives the best results, resolving physical 123 J Sci Comput 5.0 4.5 4.0 Density 3.5 3.0 Exact 2.5 λ =h1/6 λ =h1/4 λ =h1/2 λ =h2/3 λ =h3/3 2.0 1.5 1.0 0.5 -5 -4 -3 -2 -1 0 1 2 3 4 5 x (a) 4.8 4.4 4.0 Density 3.6 3.2 Exact λ =h1/6 λ =h1/4 λ =h1/2 λ =h2/3 λ =h3/3 2.8 2.4 2.0 1.6 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 x (b) Fig. 9 a Comparison of the five choices of λ. b Comparison of the solutions in the high-frequency waves. Numerical solution of the Shu–Osher problem by the WENO-P+3 scheme with different λ extrema at x = 0.647, 0.748, 0.777 and the contact discontinuities at x = 0.594 and 0.765 better. The results in Sects. 4.1.5, 4.2.3, 4.2.4, and 4.2.5 indicate that λ = h 1/6 is a good choice considering stability and numerical resolution, at least in the above four typical tested problems. This is the reason for determining the parameter λ to be h 1/6 , and all the numerical tests in the present paper also shows that the WENO-P+3 scheme with the parameter λ = h 1/6 outperforms better. 123 J Sci Comput 1.8 1.6 Density 1.4 Exact WENO-JS5 WENO-P+3 WENO-P3 WENO-N3 WENO-JS3 1.2 1.0 0.8 -5 -4 -3 -2 -1 0 1 2 3 4 5 x (a) 1.7 Density 1.6 1.5 1.4 1.3 -2.5 Exact WENO-JS5 WENO-P+3 WENO-P3 WENO-N3 WENO-JS3 -2.0 -1.5 -1.0 x (b) Fig. 10 Titarev–Toro problem: a density distribution and b the enlarged portion of the high frequency waves region 4.3 Two-dimensional Euler Systems In this subsection, we apply the proposed scheme to 2D problem for two-dimensional gas dynamics, Rayleigh–Taylor instability and Double Mach reflection of a strong shock in Cartesian coordinates. The governing two-dimensional compressible Euler equations are given by ∂U ∂F ∂G + + =0 ∂t ∂x ∂y (64) 123 J Sci Comput 1.8 1.7 1.6 1.5 Density 1.4 1.3 Exact λ = h1/6 λ = h1/4 λ = h1/2 λ = h2/3 λ = h3/3 1.2 1.1 1.0 0.9 0.8 -5 -4 -3 -2 -1 0 1 2 3 4 5 x (a) 1.7 Density 1.6 1.5 Exact λ = h1/6 λ = h1/4 λ = h1/2 λ = h2/3 λ = h3/3 1.4 1.3 -2.5 -2.0 -1.5 -1.0 x (b) Fig. 11 a Comparison of the five choices of λ. b Comparison of the solutions in the high-frequency waves. Numerical solution of the Titarev–Toro problem by the WENO-P+3 scheme with different λ where ⎞ ⎛ ⎞ ⎛ ⎞ ρu ρv ρ ⎜ ρu 2 + p ⎟ ⎜ ρvu ⎟ ⎜ ρu ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ U =⎜ ⎝ ρv ⎠ F = ⎝ ρuv ⎠ G = ⎝ ρv 2 + p ⎠ E u (E + p) v (E + p) ⎛ The total energy E and pressure p is defined by 1 1 p = (γ − 1) E − ρu 2 − ρv 2 2 2 123 (65) (66) J Sci Comput 7 6 Exact WENO-JS3 WENO-N3 WENO-P3 WENO-JS5 WENO-P+3 Density 5 4 3 2 1 0 0.0 0.2 0.4 0.6 0.8 1.0 x (a) 6.6 Exact WENO-JS3 WENO-N3 WENO-P3 WENO-JS5 WENO-P+3 6.0 5.4 Density 4.8 4.2 3.6 3.0 2.4 1.8 0.60 0.64 0.68 0.72 0.76 0.80 0.84 x (b) Fig. 12 Interacting blast wave problem: a density distribution and b the enlarged portion of the uppermost part where ρ, u, v are the density, x-velocity, y-velocity, respectively. γ is the ratio of specific heats. 4.3.1 2D Riemann Problem The test case 3 of the two-dimensional Riemann problems introduced by Lax and Liu [11] is chosen here. This problem was also simulated in [7,20] to assess the resolution of improved 123 J Sci Comput 7 6 Exact λ = h1/6 λ = h1/4 λ = h1/2 λ = h2/3 λ = h3/3 5 Density 4 3 2 1 0 0.0 0.2 0.4 0.6 0.8 1.0 x (a) 6.6 Exact λ = h1/6 λ = h1/4 λ = h1/2 λ = h2/3 λ = h3/3 6.0 5.4 Density 4.8 4.2 3.6 3.0 2.4 1.8 0.60 0.64 0.68 0.72 0.76 0.80 0.84 x (b) Fig. 13 Numerical solution of the Interacting blast wave problem by the WENO-P+3 scheme with different λ. a Comparison of the five choices of λ. b Comparison of the solutions in the upper part schemes. The problem is solved on the square domain [0, 1] × [0, 1], which is divided into four quadrants with initial constant states by lines x = 0.8 and y = 0.8: (ρ, u, v, p) = (1.5, 0, 0, 1.5) , (ρ, u, v, p) = (0.5323, 1.206, 0, 0.3) , (ρ, u, v, p) = (0.138, 1.206, 1.206, 0.029) , (ρ, u, v, p) = (0.5323, 0, 1.206, 0.3) , 123 0.8 ≤ x ≤ 1, 0.8 ≤ y ≤ 1 0 ≤ x < 0.8, 0.8 ≤ y ≤ 1 0 ≤ x < 0.8, 0 ≤ y < 0.8 0.8 < x ≤ 1, 0 ≤ y < 0.8 (67) J Sci Comput Fig. 14 Density contours of two-dimensional Riemann problem computed by the a WENO-JS3, b WENOZ3, c WENO-N3, d WENO-JS5, e WENO-Z5, f WENO-Z+5, g WENO-P3, h WENO-P+3 (λ = h 1/2 ), i WENO-P+3 (λ = h 1/4 ), and j WENO-P+3 (λ = h 1/6 ) schemes at t = 0.8. 40 contours were fit between the range of 0.14 to 1.7 123 J Sci Comput Fig. 14 continued The ratio of specific heats is set to be γ = 1.4. The final simulation time is t = 0.8 with 960 × 960 uniform grid points. The results of WENO-JS3, WENO-Z3, WENO-N3, WENOJS5, WENO-Z5, WENO-Z+5, WENO-P3, and WENO-P+3 with three different values of the parameter λ: λ = h 1/2 , λ = h 1/4 , λ = h 1/6 are displayed in Fig. 14. An examination of these results reveals that WENO-P+3 with parameter λ = h 1/6 presents a higher resolution of the structure of discrete vortex along the slip line than the WENO-JS3, WENO-Z3, WENO-N3 and WENO-JS5 schemes. 4.3.2 Two-Dimensional Rayleigh–Taylor Instability This problem has been simulated extensively in the literature [1,6,15,17,20] to assess the numerical dissipations of the numerical schemes. It happens on an interface instability between two fluid of different densities when an acceleration is directed from heavier fluid 123 J Sci Comput Fig. 15 Density contours of the Rayleigh–Taylor instability computed by the a WENO-JS3, b WENO-Z3, c WENO-N3, d WENO-JS5, e WENO-Z5, f WENO-Z+5, g WENO-P3, h WENO-P+3 (λ = h 1/2 ), i WENOP+3 (λ = h 1/4 ), and j WENO-P+3 (λ = h 1/6 ) schemes, and 15 contours were fit between the range of 0.952269 to 2.14589 123 J Sci Comput Fig. 15 continued to lighter one. The computational domain is [0, 0.25] × [0, 1], and the initial conditions are √ 2, 0, − 0.025√γ p/ρ cos (8π x) , 2y + 1 , 0 ≤ y < 0.5 (68) (ρ, u, v, p) = 1, 0, − 0.025 γ p/ρ cos (8π x) , y + 1.5 , 0.5 ≤ y < 1 with the ratio of specific heats γ = 5/3. The gravity effect is introduced by adding ρ and ρv to the right hand side of the y-momentum equation and the energy equation, respectively. Reflective boundary conditions are imposed for the left and right boundaries, and the top boundary are set as ( ρ, u, v, p) = (1, 0, 0, 2.5) and ( ρ, u, v, p) = (2, 0, 0, 1) are set to be the bottom boundary condition. The final simulation time is t = 1.95. The density contours of the solutions computed on the 240 × 960 uniform grid in Fig. 15 show than the WENO-P3 scheme proposed performs better than the WENO-N3, WENO-Z3, and WENO-JS3 scheme. It is obvious that the contact discontinuity resolution (manifested by the small scale structures) is significantly improved by the WENO-P+3 scheme. The WENO-P+3 scheme with the parameter λ = h 1/6 resolves the most complex structures even better than the WENO-JS5 scheme. 4.3.3 Double Mach Reflection of a Strong Shock The final test considered in this paper is the two dimensional double Mach reflection of a shock from an oblique surface. It describes the reflection of a planar Mach shock in air hitting 123 J Sci Comput Fig. 16 Double Mach reflection computed by the a WENO-JS3, b WENO-Z3, c WENO-N3, d WENO-JS5, e WENO-Z5, f WENO-Z+5, g WENO-P3, h WENO-P+3 (λ = h 1/2 ), i WENO-P+3 (λ = h 1/4 ), and j WENO-P+3 (λ = h 1/6 ) schemes from the top to bottom at t = 0.2 123 J Sci Comput Fig. 16 continued 123 J Sci Comput Fig. 16 continued a wedge which was initially proposed and studied in detail by Woodward and Colella [18]. This test is widely used to verify the performance of numerical methods [6,7,17,20]. It was calculated on [0, 4] × [0, 1] and the initial conditions are (8.0, 7.145, −4.125, 116.5) , x < 16 + √y 3 (69) (ρ, u, v, p) = x ≥ 16 + √y (1.4, 0, 0, 1) , 3 with a Mach 10 shock reflected from the wall with an incidence angle of 60◦ . We solve it up to time t = 0.2 using 1920 × 480 grid points with γ = 1.4. The results in [0, 3] × [0, 1] are displayed. The density contours with 40 equally spaced contour lines from ρ = 1.8 to ρ = 20 are drawn. The numerical results of the WENO-P+3 scheme with three different values of the parameter λ: λ = h 1/2 , λ = h 1/4 , λ = h 1/6 are compared with WENO-JS3, WENO-Z3, WENO-N3, WENO-JS5, WENO-Z5, WENO-Z+5 and WENO-P3 in Fig. 16. Figure 17 shows the details at the Mach stem of the density variable at the final time. It can be concluded from the small vortices captured along the slip line that the WENOP+3 scheme with the parameter λ = h 1/6 has the highest resolution among the WENOJS3, WENO-Z3, WENO-N3, and WENO-P3 schemes, even better than the WENO-JS5 scheme. 123 J Sci Comput Fig. 17 Density contours of double Mach stems on [2, 3] × [0, 1] computed by the a WENO-JS3, b WENOZ3, c WENO-N3, d WENO-JS5, e WENO-Z5, f WENO-Z+5, g WENO-P3, h WENO-P+3 (λ = h 1/2 ), i WENO-P+3 (λ = h 1/4 ), and j WENO-P+3 (λ = h 1/6 ) schemes at t = 0.2 123 J Sci Comput Fig. 17 continued 5 Conclusions In this paper, we introduce a modified third-order WENO scheme to approximate solutions of nonlinear hyperbolic conservation laws. The motivation for this study is that at relatively coarse grids, increasing the weights of less-smooth stencils is the most relevant cause than the order of accuracy at critical points for better resolution of waves. WENO-P3 scheme is constructed by slightly modified the reference smoothness indicator of WENO-N3 scheme. WENO-P+3 scheme is obtained through adding a new term to the weights of WENO-P3 scheme to further slightly increase the weight of less-smooth stencil. It is observed that the improved WENO-P+3 scheme was able to achieve significantly better results compared with WENO-JS3, WENO-Z3, WENO-N3 and WENO-P3 schemes, which is verified by several benchmark problems of one- and two-dimensional Euler system of equations. Acknowledgements The authors acknowledge the support of National Defense Fundamental Research Project (B1420133057), National Natural Science Foundation of China (51409202 and 11502180) and the Fundamental Research Funds for the Central Universities (2016-YB-016). 123 J Sci Comput References 1. Acker, F., Borges, R., Costa, B.: An improved WENO-Z scheme. J. Comput. Phys. 313, 726–753 (2016) 2. 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