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An improved WENOZ scheme with symmetrypreserving mapping
Advances in Aerodynamics volume 2, Article number: 18 (2020)
Abstract
Since the classical weighted essentially nonoscillatory (WENO) scheme is proposed, various improved versions have been developed, and a typical one is the WENOZ scheme. Although better resolution is achieved, it is shown in this article that, the result of WENOZ scheme suffers evident distortion in the longtime simulation of the linear advection equation. In order to fix the problem of WENOZ, a symmetrypreserving mapping method is proposed in this article. In the original mapping method, the weight of each substencil is used to map, which is demonstrated to cause asymmetric improvement about a discontinuity. This asymmetric improvement will lead to a distorted solution, more severe with longer output time. In the symmetrypreserving mapping method, a new variable related to the smoothness indicator is selected to map, which has the same ideal value for each substencil. Using the new mapping method can not only fix the distortion problem of WENOZ, but also improve the numerical resolution. Several benchmark problems are conducted to show the improved performance of the resultant scheme.
1 Introduction
Weighted essentially nonoscillatory (WENO) finite difference/volume scheme has become one of the most popular shockcapturing schemes to solve hyperbolic conservation laws, which are developed from the essentially nonoscillatory (ENO) schemes [1,2,3]. Unlike linear schemes using a single fixed stencil to interpolate, an r thorder ENO scheme chooses r stencils as candidates. The smoothest stencil is selected from r candidates with the aid of smoothness indicator. It is reasonable to abandon discontinuous stencils when there is a discontinuity, however, in the smooth region, r stencils together can reach (2r − 1) thorder accurate interpolation. Therefore, the ENO schemes do not make full use of all stencils’ information in the smooth region, leading to the problem of accuracy losing and computation wasting.
In order to utilize the information of lesssmooth stencils in ENO schemes, Liu et al. [4] introduced the WENO schemes. Using the smoothness indicator proposed by Liu et al., WENO scheme adopts a convex combination of r stencils, and an r thorder ENO scheme can be converted to an (r + 1) thorder WENO scheme. Jiang and Shu [5] redefined the smoothness indicator of stencils, so that an r thorder ENO scheme can be converted to an (2r − 1) thorder WENO scheme, which is known as the WENOJS scheme. However, Henrick et al. [6] pointed out that the actual accuracy order of WENOJS scheme is less than the optimal one in smooth region with critical points. To recover the accuracy at critical points, they introduced an extra mapping process where the weights of stencils are mapped to more accurate values, and the resultant scheme is known as WENOM. Then, Feng et al. [7, 8] further improved the WENOM scheme by optimizing the mapping function involved in the mapping process. To increase the relevance of lesssmooth stencils, Borges et al. [9] took both local and global smoothness indicators into consideration, and the resolution of scheme in both discontinuous and smooth regions is significantly improved. Castro et al. [10] then extended this idea to higher order schemes, which is known as the WENOZ. From another perspective, Fedkiw et al. [11, 12] observed that a small parameter ε used in formula to avoid the denominator being 0, has evident influence on the accuracy of WENO scheme. Shen et al. [13] investigated the effect of ε on the convergence property of WENO scheme in detail, and their results showed that ε = 10^{−2} can significantly improve the numerical solution for subsonic and transonic flows. Don et al. [14, 15] theoretically and numerically demonstrated that if ε is taken as a power function of the uniform grid space, i.e. ε = Ω(Δx^{m}), the WENOJS and WENOZ schemes can retain the optimal order in the smooth region, regardless of critical points. However, a larger ε leads to a less stable scheme, and thus the effect of adjusting ε to recover accuracy is problemdependent. Acker et al. [16] pointed out that to improve the resolution on a coarse grid, increasing the relevance of lesssmooth stencils is much more important than recovering accuracy at a critical point. By adding a new term that increases the relevance of lesssmooth stencils to the original formula of weight, they proposed an improved version of the fifthorder WENOZ scheme. Besides WENOM and WENOZ, other variants of WENO scheme have emerged and gained attention. Using a detector as weighting function to couple WENO reconstruction and linear compact scheme, hybrid method was proposed in [17, 18] to obtain lowdissipation and lowdispersion scheme for turbulence and aeroacoustics. Then, Sun et al. [19, 20] developed a linear minimum dispersion and controllable dissipation scheme to hybridize with WENO scheme to retain good performance in flow with shocks. However, hybrid function is critical in hybrid methods and still a big challenge. To reduce the dissipation, Martin et al. [21] proposed WENOSYMBO scheme through adding a downwind stencil and optimizing the ideal coefficients for better spectral property. Hu et al. [22] proposed a new weighting strategy and new smoothness indicator for the downwind stencil, leading to the adaptive centralupwind sixthorder WENO scheme (WENOCU6). However, since the background central scheme cannot suppress disturbances from accumulated dispersion errors, spurious waves may appear at marginallyresolved wave numbers. Based on Lagrangian interpolation polynomial, Pan et al. [23, 24] proposed a new smoothness indicator with more succinct form compared to the classical one by Jiang and Shu [5], leading to a scheme named WENO η. Don et al. [25] found that due to the numerical unstable form of the local smoothness indicators, asymmetry occurs in an otherwise problem for seventh and higherorder WENO schemes. Then, they introduced an equivalent but compact and numerical stable form of the smoothness indicators. Recently, Fu et al. [26,27,28] proposed a family of targeted ENO (TENO) schemes. By using a set of loworder candidate stencils with incrementally increasing width, discontinuities and smallscale fluctuations are efficiently separated, and the numerical dissipation is significantly diminished by an ENOlike stencil selection.
Due to better resolution while with almost the same computational cost as WENOJS, WENOZ has gained extensive attention. In this article, from the simulation of linear advection equation, it is observed that the numerical solution of WENOZ suffers distortion problem near a discontinuity. To overcome this deficiency, a symmetrypreserving mapping method is proposed to improve the WENOZ scheme. This article is organized as follows. Section 2 describes the classical WENOJS scheme and WENOZ scheme. In Section 3, the original mapping method and the symmetrypreserving mapping method are introduced in detail. The mapped WENOZ is proposed in Section 4. Numerical experiments with one and two dimensional benchmark problems are presented in Section 5. Conclusions are drawn in Section 6.
2 WENO scheme
2.1 WENO reconstruction
Here, a grid with uniform space Δx is used, and the positions of cell center and cell boundary are represented as x_{i} = iΔx, \( {x}_{i+1/2}={x}_i+\frac{\varDelta x}{2},i=0,\dots, N \), respectively. The values of flux at cell center and cell boundary are denoted as f_{i} = f(x_{i}) and f_{i + 1/2} = f(x_{i + 1/2}). The numerical flux h(x) is defined as
so that \( \frac{1}{\varDelta x}\left({h}_{i+1/2}{h}_{i1/2}\right) \) is exactly equal to the firstorder derivative of f at cell center i, i.e. \( {f}_i^{\hbox{'}} \).
Without loss of generality, the fifthorder WENO (WENO5) scheme is taken as an example to illustrate the WENO reconstruction procedure. In linear scheme, a fifthorder approximation of h(x) only needs a fivepoint stencil S^{5}, while in WENO5, S^{5} is divided into three substencils {S_{0}, S_{1}, S_{2}}, and each substencil contains three adjacent grid cells, shown in Fig. 1. In the smooth region, WENO5 combines each substencil with corresponding ideal weights to obtain a fifthorder approximation of h(x). While in the region with discontinuities, to avoid numerical oscillation, substencils containing the discontinuity are suppressed by assigning small weights to them.
Let \( \hat{h} \) denote the approximation of h(x), and the three substencils of WENO5 give
where the superscript is used to distinguish different substencils. \( {h}_{i1/2}^k \) can be obtained via shifting each index in \( {h}_{i+1/2}^k \) by −1.Then the \( {\hat{h}}_{i\pm 1/2} \) of stencil S^{5} is obtained by a weighted average of \( {h}_{i\pm 1/2}^k \),
In the smooth region, the weights approach their ideal values, which are
2.2 WENO5JS scheme
In the classical fifthorder WENO (WENO5JS), proposed by Jiang and Shu [5], the nonlinear weights are defined as
ε is a small positive parameter to avoid denominator being zero, and the power parameter p ≥ 1 controls the amount of dissipation, usually taken as 2. β_{k} is the smoothness indicator of kth substencil, defined as
In the smooth region, in order to achieve optimal order of accuracy, i.e., satisfying the relation
the weights of substencils need to meet certain constraints. A strong sufficient condition is used in [6], given as
and a weaker sufficient condition is derived in [14],
Taylor series expansions of Eq.(6) at cell center x_{i} give

If f^{'} ≠ 0, one can have

If f^{'} = 0 and f^{''} ≠ 0 (critical point), one can have
If β_{k} in Eq.(6) takes the form of
where D is a nonzero constant independent of substencils, and m is a positive parameter. Then Eqs.(5) and (13) give
and
which lead to
In the context of this article, a critical point refers to the point where the firstorder derivative is zero while the secondorder derivative is nonzero. From Eqs.(11) and (12), two observations can be obtained in the smooth region as follows.

At a point which is not a critical point, ω_{k} obtained by WENO5JS satisfies the weaker sufficient condition for achieving optimal order, i.e. Eq.(9).

At a critical point, ω_{k} obtained by WENO5JS only satisfies ω_{k} = d_{k} + Ο(Δx), resulting in the accuracy loss at this point.
2.3 WENO5Z scheme
Borges et al. [9] found that the increase of weights of lesssmooth substencils can substantially improve the numerical solution near both discontinuities and smallscale structures. Based on this, through the novel use of higher information that is already presented in the framework of classical WENO5JS, they devised new smoothness indicators given as
where τ_{5} = ∣ β_{0} − β_{2}∣ is the higher order smoothness indicator. τ_{5} has two important properties, which are

If S^{5} is smooth, then τ_{5} = Ο(Δx^{5}).

If S^{5} is discontinuous, then τ_{5} = Ο(1) ≤ max(β_{0}, β_{1}, β_{2}).
Substitute Eq.(17) into Eq.(5), and the new weights are obtained by
which lead to the fifthorder WENOZ (WENO5Z). At a critical point with f^{'} = 0 and f^{''} ≠ 0, β_{k} = Ο(Δx^{4}) in Eq.(10), leading to \( \frac{\tau_5}{\beta_k+\varepsilon }=\mathrm{O}\left(\varDelta x\right) \) and thus ω_{k} = d_{k} + Ο(Δx). Therefore, Eq.(18) does not solve the problem of accuracy loss at a critical point. However, due to the larger assignment of weights to the lesssmooth substencils, the numerical results of WENO5Z are substantially improved compared to WENO5JS, and slightly better compared to WENO5M, details shown in [9].
Castro et al. [10] then fixed the problem of accuracy loss and extended WENO5Z to higher order. For (2r − 1) thorder WENOZ, the weights are obtained by
The power parameter p ≥ 1 in Eq.(19) plays two roles. On one hand, p controls the amount of numerical dissipation, and the scheme is more dissipative with a larger p. On the other hand, the accuracy at a critical point can be recovered by increasing p. The former is intuitive and easy to understand. As for the latter, at a critical point, \( {\left(\frac{\tau_5}{\beta_k+\varepsilon}\right)}^p=\mathrm{O}\left(\varDelta {x}^p\right) \) leads to ω_{k} = d_{k} + Ο(Δx^{p}), and thus the weaker sufficient condition, Eq.(9), for achieving optimal order is satisfied if p ≥ 2. Changing the value of p to alter convergence rate at critical points is a unique feature of WENOZ, while in WENOJS and WENOM, p only controls the amount of numerical dissipation. In order to balance accuracy and stability, for (2r − 1) thorder scheme, p is generally suggested to be r − 1 [10].
3 The symmetrypreserving mapping method
3.1 The mapping method by Henrick
Henrick et al. [6] first pointed out the fact that WENO5JS suffers accuracy loss at a critical point, and then they introduced an extra mapping process to recover the accuracy, where the weights are mapped to more accurate values. To be specific, the mapping is achieved through a set of mapping functions having the form of
The function is smooth and has the following features: g_{k}(0) = 0, g_{k}(1) = 1, g_{k}(d_{k}) = d_{k}, \( {g}_k^{\hbox{'}}\left({d}_k\right)=0 \) and \( {g}_k^{\hbox{'}\hbox{'}}\left({d}_k\right)=0 \). Once the weight of WENO5JS in Eq.(5) is obtained, the new unnormalized weight is given by
Taylor series expansion of Eq.(21) gives
Then, new weight is obtained by
Substitute Eq.(22) into Eq.(23), and new weight satisfies
Eq.(24) indicates that the strong sufficient condition, Eq.(8), is satisfied. Therefore, the problem of accuracy loss at a critical point is fixed. The resultant mapped WENO scheme is generally abbreviated as WENOM.
3.2 The asymmetric improvement of WENO5M
Here, the linear advection equation,
with periodic boundary conditions, is taken as an example for illustration. Consider a single square wave, and the initial condition is
The thirdorder TVD RungeKutta method is used to do the discretization in time, expressed as
Time is integrated to t = 20, and the numerical results are shown in Fig. 2. Due to the less dissipation of WENO5M compared to WENO5JS, an improvement in the result near the discontinuity is observed in Fig. 2 (a). From the absolute pointwise errors in Fig. 2 (b), however, the improvement on both sides of the discontinuity is not in a symmetric fashion. Specifically, the error curve of WENO5JS is almost symmetric about the discontinuity, shown with a red line in Fig. 2 (b), while WENO5M has obvious larger error on the right side of the discontinuity, shown with a blue line in Fig. 2 (b). Consequently, as is shown in Fig. 2 (a), the improvement of WENO5M on the right side of the discontinuity is obviously less than that on the left side. Due to the asymmetry near the discontinuity, the whole error curve of WENO5M is no longer symmetric about the center of domain x = 0, shown in Fig. 2 (b).
In order to clarify the cause of this asymmetric improvement of WENO5M, a simple model shown in Fig. 3 is used. On a sixpoint domain with a discontinuity at the center, there are two sets of fivepoint stencils symmetric about the discontinuity. A quantity with superscript L belongs to the left stencil, while with superscript R belongs to the right stencil.
According to Eq.(6) and due to the symmetry, smoothness indicators of left and right stencils have the following relation,
Obviously, for the left stencil, 0th substencil is the smoothest, and 2th substencil is the least smooth. While for the right stencil, 2th substencil is the smoothest, and 0th substencil is the least smooth.
According to Eq.(5), the weight of each substencil on the same stencil satisfies the relation,
Assuming that θ, σ are weights assigned to the leastsmooth substencils of left and right stencils, the weight of each substencil can be written as
WENO scheme behaves like an ENO scheme at a discontinuity, and thus θ, σ are small values close to 0. Then, new weights are obtained through the mapping process, i.e.,
As θ, σ approach 0, it is approximately obtained that
From Eq.(32), it is observed that after mapping, the magnification of weight of the leastsmooth substencil is not identical for both sides of the discontinuity, i.e. \( \frac{\omega_2^{\mathrm{L},\mathrm{M}}}{\omega_2^{\mathrm{L}}}\ne \frac{\omega_0^{\mathrm{R},\mathrm{M}}}{\omega_0^{\mathrm{R}}} \), obviously larger on the right side. It is the mismatched magnification that results in the asymmetric improvement of WENO5M around a discontinuity.
3.3 A symmetrypreserving mapping method
It is the derivative of mapping function at 0 that determines the magnification of weight of lesssmooth substencils, i.e.,
Due to the difference in ideal weights for each substencil, mapping the weight with the function by Henrick inevitably causes the problem of asymmetric improvement. There are two ways to overcome the problem of mismatched magnification. One way is to redesign a set of mapping functions with identical derivatives at 0, and the other is to choose a new mapping object that has the same ideal value for each substencil. An example of the former is the work in [7], where the new mapping function is smooth and piecewise,
where
and
The new function has identical derivative at 0,
In [7], the researchers noticed the distortion problem of WENO5M, but they improperly attributed this to the numerical instability, which was thought to be caused by overamplification of weight of lesssmooth substencil. Therefore, they proposed the function with derivative of 0 at 0, in the form of Eq.(34), to retain the ENO property at a strong discontinuity. Although the reason is not properly understood, due to restricting all derivatives to be identical at 0, using mapping function in the form of Eq.(34) is able to avoid the problem of asymmetric improvement. However, extra constraint on derivative brings difficulty in designing concise mapping function, and complicated function will result in significantly increased computational cost. The other way is choosing a new mapping object that has identical ideal value for each substencil, so that extra constraint on derivative at 0 is not necessary. Furthermore, when the ideal value is the same, the mapping function is unique, and the function design is much simplified. Therefore, the second way is more reasonable.
In the formula of weight Eq.(5), besides the weight ω_{k}, there is another quantity related to each substencil, i.e. the smoothness indicator β_{k}. In the smooth region, different from the weight ω_{k}, the smoothness indicator β_{k} is equal for each substencil, and thus its ideal value is unique. In order to keep proportional to ω_{k}, consider the inverse of smoothness indicator with power parameter p, i.e. \( \frac{1}{{\left({\beta}_k+\varepsilon \right)}^p} \). But for mapping, it is necessary to normalize them to a limited interval,
The normalized variable, λ_{k}, is the new mapping object, and its unique ideal value is
In general, the ideal value of λ_{k} for (2r − 1) thorder WENO scheme is \( \frac{1}{r} \).
Substitute Eq.(12) into Eq.(38), and it is obtained at a critical point that
It is straightforward to apply the mapping function, originally designed for ω_{k}, to do the mapping for λ_{k}. Considering the function by Henrick, the mapping function for λ_{k} is unique and written as
Similar to Eq.(22), after mapping, it is obtained that
Then, new weight is evaluated and satisfies
Eqs.(43) and (24) indicate that at a critical point, mapping λ_{k} can achieve the same effect as mapping ω_{k}.
According to Eqs.(28) and (38), λ_{k} for the left and right stencils in Fig. 3 satisfies
After mapping with function Eq.(41), the mapped value still satisfies
According to Eq.(43), it is obtained that
Based on Eqs.(45) and (46), one has
As \( {\omega}_2^{\mathrm{L}},{\omega}_2^{\mathrm{R}} \) approach 0, \( \frac{\omega_0^{\mathrm{L},\mathrm{FM}}}{\omega_0^{\mathrm{L}}} \) and \( \frac{\omega_2^{\mathrm{R},\mathrm{FM}}}{\omega_2^{\mathrm{R}}} \) approach 1. Therefore, it is approximately obtained that
From Eqs.(28) and (29), one has
According to Eqs.(33), (47), (48) and (49), it is obtained that
Eq.(50) indicates that using the new mapping object λ_{k}, the problem of mismatched magnification of weights of lesssmooth substencils is fixed.
In this article, WENO5JS with the symmetrypreserving mapping method is abbreviated as WENO5FM. As is shown in Fig. 2 (b), the error curve of WENO5FM is symmetric about the discontinuity, and almost symmetric about the center x = 0 on the whole domain. Consequently, compared to WENO5JS, the improvement of WENO5FM on both sides of a discontinuity is in a symmetric fashion. To present an intuitive comparison, the linear advection equation initialized by Eq.(26), is still numerically tested here. Results of the first way, where Eq.(34) is used to map ω_{k}, are also presented, n = 6 as suggested in [7]. The corresponding scheme is abbreviated as WENO5MP. As is shown in Fig. 4, with longer output time t = 100, the asymmetric improvement is more evident compared the case of t = 20. On the left side of the discontinuity, three schemes have similar results, while on the right side, the result of WENO5M is much worse than the others and suffers obvious distortion.
It is observed from Fig. 5 that at t = 250, both sides of the discontinuity are influenced in the result of WENO5M, and similar to the right side of the discontinuity, the result on the left side also becomes distorted. On the right side of the discontinuity, the results of WENO5MP and WENO5FM are still close to each other, similar to the case of t = 100. However, on the left side, there is obvious overshoot in the result of WENO5MP, shown with a blue line in Fig. 5. Taking the result at t = 400 in Fig. 6 into consideration together, it can be inferred that the overshoot of WENO5MP is caused by numerical instability. From the results at t = 100, 250, 400, WENO5FM is always able to obtain nonoscillatory results and the improvement is symmetric about the discontinuity.
As is mentioned before, to enhance the numerical stability near a discontinuity, the function used in WENO5MP constrains the derivative at 0 to be 0, but numerical oscillation is observed in Figs. 5 and 6. To figure out the numerical instability, it is necessary to have an observation at the distribution of mapping functions. As is shown in Fig. 7, the black dashed line corresponds to the case without mapping, while the red and blue lines are related to functions used in WENO5FM and WENO5MP, respectively. Due to different ideal weights for each substencil, there are three different functions involved in WENO5MP. Only the function for the minimum weight \( {d}_0=\frac{1}{10} \), is presented in Fig. 7.
It is observed that the magnification of weight near 0 is smaller with blue line, which is beneficial to capture strong discontinuities. However, for a weak discontinuity in numerical sense, i.e., the weight is not so close to 0 or 1 but deviates significantly from the ideal value, the adjustment of weight is obviously stronger with blue line. From the results in Figs. 4, 5 and 6, it can be concluded that even with larger magnification of weight near strong discontinuity, the numerical stability of WENO5FM is still ensured. By contrast, there is no oscillation observed in the result of WENO5MP at t = 100, while evident oscillation occurs at t = 250 when the initial strong discontinuity is smoothed out and becomes much weaker due to numerical dissipation. It can be inferred that, the numerical instability of WENO5MP comes from the overadjustment of weight near a weak discontinuity. As is mentioned previously, there is a controllable parameter n in the function defined by Eq.(34). Functions with n = 4 and n = 2 are plotted in Fig. 7, and it is shown that with smaller n, the adjustment of weight is weaker, especially in the region with weak discontinuities. Figure 8 shows the results of WENO5MP with n = 6, 4, 2. In the results of n = 4, 2, there is no numerical oscillation observed, while n = 2 leads to a slightly more dissipative result. This further supports the inference that the numerical instability is due to the overadjustment of weight near a weak discontinuity.
The above results suggest that weak discontinuity is also important for numerical stability. To ensure a stable mapping, excessive adjustment should be avoided near both strong and weak discontinuities. The core of mapping method lies in the mapping function, and different requirements on a scheme, such as strong stability, low dissipation, can be met with specifically designed function. In terms of function design, using λ_{k} to map has two main advantages over using ω_{k}. On one hand, for (2r − 1) thorder scheme, there are r different ideal weights and thus r different functions need to be designed. While there is only one function to be designed for using λ_{k} to map due to the unique ideal value, which greatly simplifies the function design. On the other hand, the minimum ideal weight is closer to 0 with higher order, e.g., \( {d}_0=\frac{1}{10},\frac{1}{35},\frac{1}{126} \) for fifth, seventh and ninthorder schemes, respectively. Ideal value close to 0 brings difficulty in balancing accuracy recovery and numerical stability. By contrast, \( \overline{\lambda}=\frac{1}{3},\frac{1}{4},\frac{1}{5} \) for fifth, seventh and ninthorder schemes respectively, which significantly alleviates this dilemma.
4 Improved WENO5Z with mapping
4.1 The distortion of WENOZ in longtime simulation
Figure 9 shows the numerical results of WENO5JS in simulating linear advection equation initialized by Eq.(26). Due to the role of numerical dissipation, the sharp discontinuity is captured in the form of smooth curve at a short time t = 4. However, with longer output time t = 100, 250, the result near a discontinuity suffers not only greater dissipation, but also distortion, obviously shown with a green line at t = 250. Greater dissipation with longer output time is normal, while the distortion is not expected and severely deviates the numerical solution from the real solution.
By introducing a global smoothness indicator, the relevance of lesssmooth substencil is increased in WENO5Z, and thus less dissipation is achieved compared to WENO5JS. However, it is shown in Fig. 10 that the distortion still exists in the results of WENO5Z. Different from WENO5JS, the distortion is already obvious at t = 100 for WENO5Z, shown with a blue line in Fig. 10, while the distortion at t = 250 is less severe compared to WENO5JS.
The distortion problem of WENO5JS and WENO5Z, in the longtime simulation of linear advection equation, is also observed by Feng et al. [7]. By designing a new set of mapping functions that satisfy \( {g}_k^{\hbox{'}}(0)=0 \), the distortion problem of WENO5JS was fixed with an extra mapping process. Inspired by this, the mapping method can also be potential to overcome the distortion problem of WENO5Z. However, based on the results in Section 3, it is necessary to adopt the symmetricpreserving mapping method.
4.2 Mapped WENO5Z scheme
It is straightforward to do the symmetricpreserving mapping for WENO5Z. Specifically, the new variable for WENO5Z to map is
The ideal value of \( {\lambda}_k^{\mathrm{Z}} \) is the same as \( \overline{\lambda} \), i.e.,
Therefore, the mapping function is also the same as Eq.(41). Through mapping, it is obtained that
Then, the new weight is obtained by
In this article, WENO5Z with the symmetricpreserving mapping method is abbreviated as WENO5ZM.
To demonstrate the effect of the mapping on WENO5Z, the same numerical simulation in Section 4.1 is conducted. As is shown in Fig. 11, there is no more distortion observed in the results of WENO5ZM at t = 100, 250, shown with the green and orange lines. With distortion problem fixed, the improvement near the discontinuity is significant compared to WENO5Z.
To further demonstrate the performance of WENO5ZM in problems with multiple discontinuities, consider the following initial condition,
where \( G\left(x,\beta, z\right)={e}^{\beta {\left(xz\right)}^2}, \)\( F\left(x,\alpha, a\right)=\sqrt{\max \left(1{\alpha}^2{\left(xa\right)}^2,0\right)}, \)z = − 0.7, δ = 0.005, \( \beta =\frac{\log 2}{36{\delta}^2}, \)a = 0.5, α = 10. The initial condition consists of a Gaussian function, a square function, a piecewise linear triangle function and an ellipse function. It is shown in Fig. 12 that at a short time t = 4, there is no evident distortion in the results of WENO5JS and WENO5Z, and WENO5Z has slight improvement near a discontinuity compared to WENO5JS. By contrast, the improvement of WENO5ZM is much significant, clearly shown in the zoomedin solutions with the green line. As is shown in Figs. 13 and 14, at t = 100, 250, there is evident distortion in both results of WENO5JS and WENO5Z, and it is more severe with WENO5JS. As expected, all the results of WENO5ZM free from the distortion problem, and significantly improved results near discontinuities are observed. For comparison, the result of WENO5FM is also presented in this test case. As is shown in these figures, WENO5FM produces results very similar to that of WENO5ZM, Specifically, WENO5FM shows slightly more dissipation compared to WENO5ZM, see the zoomedin solutions.
Following the approximated dispersion relation (ADR) analysis [29], the spectral properties of various schemes are shown in Fig. 15. With the symmetrypreserving mapping, the bias of overall scheme towards the linear scheme is increased. Thus, the spectral properties of WENO5FM and WENO5ZM are improved compared to WENO5JS and WENO5Z, respectively. Since WENO5Z has better resolution than WENO5JS in the intermediatetohigh wavenumber range, WENO5ZM shows lower dispersion and dissipation than WENO5FM. Notice that WENO5Z shows better spectral properties than WENO5FM in the intermediate wavenumber range, roughly from ξ = 1 to ξ = 1.5. However, as is shown in Figs. 12, 13 and 14, WENO5FM produces much better results than WENO5Z near discontinuities.
4.3 Prediscrete mapping method
The disadvantage of mapping method is obvious, i.e., due to involving extra mapping process, the computational cost will be increased. As was reported in [10], WENO5M led to 20% ~ 30% extra cost compared to WENO5JS. More complex function, like Eq.(34), will undoubtedly worsen this disadvantage.
In order to reduce the cost of mapping process, Hong et al. [30] proposed a prediscrete mapping method. The method consists of three steps, which are described as follows.

Step 1.
Discretize the interval (0, 1), range of λ_{k}, with a uniform space \( \varDelta \lambda =\frac{1}{N} \), where N is the number of subintervals and satisfies

Step 2.
The discrete sequence of λ_{k} can be expressed as λ_{k, j} = 0 + jΔλ, j = 0, …, N, and corresponding discrete sequence of mapped value can be obtained by

Step 3.
The index of the point closest to λ_{k} is obtained by
where int denotes a rounding operation. Finally, the mapped value of λ_{k} is given by
In fact, λ_{k, index} is a firstorder approximation to λ_{k}, and in the smooth region it can be written as
Also, Taylor series expansion of g(λ_{k, index}) at \( \overline{\lambda} \) gives
Equations (61) and (42) indicate that the prediscrete mapping method can do the same mapping as the original method.
In the prediscrete mapping method, the calculation of mapping function, which is the most timeconsuming in mapping process, is only done once in step 1 before the iteration. Instead of calculating the function directly, the mapped value is obtained by one multiplication, one rounding operation and one addressing operation, which leads to lower computational cost. Moreover, this low cost is independent of the specific form of mapping function, and thus all kinds of functions can be used at the same cost, which frees the function design from the limit of computational cost.
5 Numerical results
In this section, several one and twodimensional Euler problems are conducted to validate the performance of WENO5ZM. The Roe scheme [31] is used to solve a local Riemann problem at cell boundary. An explicit thirdorder TVD RungeKutta method, defined as Eq.(27), is used to solve the resulting set of ordinary differential equations in time. Referring to [14], the parameter ε is set to be as small as 10^{−20}, and the power parameter p is taken as 2 in all schemes. All numerical experiments are computed with double precision (64 bits). In the prediscrete mapping method, N is taken as 10^{4} for all cases.
5.1 Onedimensional Euler equations
The strong conservative form of onedimensional Euler equations of gas dynamics is
where ρ, u, p, E represent the density, velocity, pressure and total energy, respectively. The equation of state,
is supplemented to close the system of equations. γ is the specific heat ratio and taken as 1.4 if not specified.
5.1.1 The Riemann problem of Sod
Sod problem [32] is used to demonstrate the ability of shockcapturing. The initial condition on domain [−0.5, 0.5] is
Zerogradient boundary conditions are imposed at x = ± 0.5, and the numerical solutions are shown in Fig. 16. The solution consists of a leftgoing rarefaction wave, a central contact discontinuity and a rightgoing shock wave. All results are nonoscillatory, and due to less dissipation, WENO5ZM shows better resolution near discontinuities than WENO5Z, clearly observed in the zoomedin solutions.
5.1.2 Mach 3 shockdensity wave interaction
The initial condition is a Mach 3 shock interacting with a perturbed density field, and given as
At x = ± 5, the zerogradient boundary conditions are applied, and the numerical solutions are shown in Fig. 17. It is observed that in regions with discontinuities and highfrequency wave, WENO5ZM shows the best resolution. While due to the excessive dissipation, the highfrequency wave is nearly smoothed out in the result of WENO5JS.
5.1.3 Interacting blast waves
The test consists of two interacting blast waves. The strong shocks in the solution are computationally hard to solve, and schemes with unstable tendencies often fail to converge in this test. The initial condition on domain [0, 1] is
Reflective boundary conditions are used at x = 0 and x = 1, and the numerical solutions are shown in Fig. 18. Obviously shown in the zoomedin solutions, the result of WENO5ZM shows the best resolution near discontinuities without any oscillation.
5.2 Twodimensional Euler equations
For twodimensional problem, the strong conservative form of Euler equations is
where y, v represent the other direction and the velocity component in this direction, respectively. The twodimensional equation of state is
5.2.1 RayleighTaylor instability problem
The RayleighTaylor instability (RTI) occurs at the interface between two fluids with different density when the heavier fluid accelerates to the lighter fluid. The computational domain is [0, 0.25] × [0, 1], and the initial condition is
where \( c=\sqrt{\gamma P/\rho } \), is the speed of sound. The interface of two fluids with different density is at y = 0.5. The specific heat ratio in Eq.(68) is specified as γ = 5/3 in this case. At x = 0 and x = 0.25, reflective boundary conditions are used. ρ = 2, u = v = 0, p = 1 are set for y = 0, while ρ = 1, u = v = 0, p = 2.5 are set for y = 1. To exert the effect of gravity, the source terms ρ, ρv are explicitly added to the right hand side of third and fourth equations of Eq.(67).
As is shown in Fig. 19, all schemes can obtain symmetric results. Due to the KelvinHelmholtz instability, vortices are rolled up around the interface indicated by clustered contour lines. The least dissipation of WENO5ZM leads to a result with the most details of vortices, while smallscale vortices are nearly smoothed out in the result of WENO5JS due to its overdissipation.
5.2.2 Double Mach reflection problem
Double Mach reflection (DMR) describes a Mach 10 moving shock with an angle of 60^{∘} with respect to the wall. The computational domain is [0, 4] × [0, 1], and the initial condition is
where x_{0} = 1/6 is the leading position of the wall, and θ = π/6 is the angle between initial oblique shock and yaxis. Supersonic inflow and outflow boundary conditions are used at x = 0 and x = 4 respectively. At the lower boundary y = 0, postshock values are used at (0, x_{0}), while a reflective boundary condition is used at (x_{0}, 4). As for the upper boundary y = 1, the exact solution of the Mach 10 moving oblique shock is applied.
It is observed from Fig. 20 that all schemes can give results without any oscillation, and the main difference is reflected near the slip line. Vortices rolled up around the slip line can be observed in the result of WENO5Z, while in the result of WENO5ZM, they are more obvious, indicating the improvement of the latter.
5.2.3 2D Riemann problem
The 2D Riemann problem [33] is defined on a square domain [0, 1] × [0, 1], which is divided into four rectangular subdomains with lines x = 0.8 and y = 0.8, and the initial condition is
Figure 21 shows the density contour obtained with WENO5JS, WENO5Z and WENO5ZM. It is generally thought that capturing more smallscale structures indicates better resolution in this problem. Therefore, from the comparison of the three results, WENO5ZM obviously has the best resolution.
5.2.4 Isentropic vortex propagation problem
The initial condition on a square domain [0, 10] × [0, 10] is given by
where κ = 5 is the vortex strength, and (x_{c}, y_{c}) = (5, 5) is the initial location of the vortex center. Periodic boundary conditions are used for all boundaries. Referring to [34], this test is tested under 10 periods and 100 periods with t = 100 and t = 1000, respectively. After 10 periods propagation with t = 100, it is shown in Fig. 23 (a) that the three results are very close to each other and in good agreement with the reference result. However, as is shown with the contours in Fig. 22, the difference becomes evident after 100 periods propagation with t = 1000. A quantitative comparison is given in Fig. 23 (b). It is observed that WENO5JS and WENO5ZM still keep the vortex at the center of the computational domain, while in the result of WENO5Z, the vortex travels a little faster and thus is off center. Moreover, the result of WENO5ZM is the closest to that of the optimal linear scheme but with slightly more dissipation.
5.3 The computational cost
Due to involving an extra mapping process, the computational cost of WENO5ZM must be higher than that of WENO5Z. If the improvement of WENO5ZM comes at the cost of significantly increased computational time, then the attractiveness of the improved scheme will be greatly reduced. In order to give an intuitive comparison, the computational cost is numerically investigated here.
Take the DMR problem in Section 5.2.2 as the object to test, and to eliminate occasionality, the experiment is repeated three times under the same conditions. The computation is performed with double precision on a 2.40GHz Intel Xeon CPU E5–2676, and the code is developed in Fortran. In average, the computational cost of WENO5ZM is about 6% higher than that of WENO5Z. Compared to 20% ~ 30% extra cost of WENO5M reported in [10], such a low extra cost indicates the efficiency of WENO5ZM.
6 Conclusions
In the original mapping method proposed by Henrick, the weight of each substencil is used to map for recovering the accuracy at critical points. The numerical experiment of linear advection equation in this article shows that, the result around a discontinuity is improved but in an asymmetric fashion. With longer output time, this asymmetric improvement results in a solution with evident distortion. From a qualitative analysis, it is found that there is mismatched magnification of weight of lesssmooth substencil for both sides of a discontinuity, due to the difference in ideal weights of substencils. To overcome the problem of asymmetric improvement, a symmetrypreserving mapping method is proposed in this article. Instead of the weight, a new variable related to the smoothness indicator is chosen as the mapping object, which has the same ideal value for each substencil. Moreover, since the function is unique and the ideal value is not close to 0, function design is much simplified in the new mapping method. In the longtime simulation of linear advection equation, it is found that the numerical solution of WENO5Z suffers evident distortion, which seriously degrades the quality of solution. When applying the new mapping method to WENO5Z, the distortion no longer exists, and thus the result is significantly improved.
Numerical experiments with one and twodimensional Euler equations are conducted to validate the performance of mapped WENO5Z. In all problems, the results of mapped WENO5Z show better resolution at both discontinuities and smallscale structures compared to WENO5Z, and there is no oscillation observed in these results. The extra computational cost, brought by the mapping process, is moderate with a value of 6%, which is numerically investigated by the DMR problem.
Availability of data and materials
The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
The authors would like to acknowledge the funding support of this research by National Natural Science Foundation of China (No. 11732013), National Numerical Windtunnel (No. NNW2019ZT3A15) and Foundation of National Key Laboratory (No. JCKYS6142201190304).
Funding
The present study is supported by National Natural Science Foundation of China (No. 11732013), National Numerical Windtunnel (No. NNW2019ZT3A15) and Foundation of National Key Laboratory (No. JCKYS6142201190304).
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ZH was responsible for the formulation of the problem, the analyses of numerical results, and the writing. ZY directed the whole work. KY carried out the numerical computation. The author(s) read and approved the final manuscript.
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Hong, Z., Ye, Z. & Ye, K. An improved WENOZ scheme with symmetrypreserving mapping. Adv. Aerodyn. 2, 18 (2020). https://doi.org/10.1186/s4277402000043w
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DOI: https://doi.org/10.1186/s4277402000043w