Towards building the OP-Mapped WENO schemes: A general methodology

A serious and ubiquitous issue in existing mapped WENO schemes is that most of them can hardly preserve high resolutions and in the meantime prevent spurious oscillations on solving hyperbolic conservation laws with long output times. Our goal in this article is to address this widely concerned problem [3, 4, 15, 29, 16, 18]. In our previous work [18], the order-preserving (OP) criterion was originally introduced and carefully used to devise a new mapped WENO scheme that performs satisfactorily in long-run simulations, and hence it was indicated that the OP criterion plays a critical role in the maintenance of low-dissipation and robustness for the mapped WENO schemes. Thus, in our present work, we firstly define the family of the mapped WENO schemes, whose mappings meet the OP criterion, as OP-Mapped WENO. Next, we attentively take a closer look at the mappings of various existing mapped WENO schemes and devise a general formula for them. It helps us to extend the OP criterion into the design of the improved mappings. Then, we propose the generalized implementation of obtaining a group of OP-Mapped WENO schemes, named MOP-WENO-X as they are developed from the existing mapped WENO-X schemes, where the notation"X"is used to identify the version of the existing mapped WENO scheme. Finally, extensive numerical experiments and comparisons with competing schemes are conducted to demonstrate the enhanced performances of the MOP-WENO-X schemes.


Introduction
The essentially non-oscillatory (ENO) schemes [8,9,10,7] and the weighted ENO (WENO) schemes [26,27,19,13,25] have developed quite successfully in recent decades to solve the hyperbolic conservation problems, especially those that may generate discontinuities and smooth small-scale structures as time evolves in its solution even if the initial condition is smooth. The main purpose of this paper is to find a general method to introduce the orderpreserving (OP) mapping proposed in our previous work [18] for improving the existing mapped WENO schemes for the approximation of the hyperbolic conservation laws in the form where u = (u 1 , · · · , u m ) ∈ R m is the vector of the conserved variables and F(u) is the vector of the Cartesian components of flux. Harten et al. [8] introduced the ENO schemes. They used the smoothest stencil from r possible candidate stencils based on the local smoothness to perform a polynomial reconstruction such that it yields high-order accuracy in smooth regions but avoids spurious oscillations at or near discontinuities. Liu, Osher, and Chan [19] introduced the first WENO scheme, an improved version of the ENO methodology with a cell-averaged approach, by using a nonlinear convex combination of all the candidate stencils to achieve a higher order of accuracy than the ENO same article, the definition of the order-preserving (OP) mapping was given and suggested as an additional criterion in the design of the mapping function. Then a new mapped WENO scheme with its mapping function satisfying the additional criterion was proposed. Extensive numerical experiments showed that this scheme can achieve the optimal convergence order of accuracy even at critical points. It also can decrease the numerical dissipations and obtain high resolution but does not generate spurious oscillation near discontinuities even if the output time is large. Moreover, it was observed clearly that it exhibits a significant advantage in reducing the post-shock oscillations when calculating the steady problems with strong shock waves in two dimension.
In this article, the idea of introducing the OP criterion into the design of the mapping functions proposed in [18] is extended to various existing mapped WENO schemes. First of all, we give a common name of OP-Mapped WENO to the family of the mapped WENO schemes whose mappings are OP. A general formula for the mapping functions of various existing mapped WENO schemes is obtained that allows the extension of the OP criterion to all existing mapped WENO schemes. The notation MOP-WENO-X is used to denote the improved mapped WENO scheme considering the OP criterion based on the existing WENO-X scheme. A new function named minDist is defined (see Definition 4 in subsection 3.3 below). A general algorithm to construct OP mappings through the existing mapping functions by using the minDist function is proposed. Extensive numerical tests are conducted to demonstrate the performances of the MOP-WENO-X schemes: (1) a series of accuracy tests shows the capacity of the MOP-WENO-X schemes to achieve the optimal convergence order in smooth regions with first-order critical points and their advantages in long output time simulations of the problems with very high-order critical points; (2) the one-dimensional linear advection equation with two kinds of initial conditions for long output times are then presented to demonstrate that the MOP-WENO-X schemes can obtain high resolution and meanwhile avoid spurious oscillation near discontinuities; (3) some benchmark tests of steady problems with strong shock waves modeled via the two-dimensional Euler equations are computed, it is clear that the MOP-WENO-X schemes also enjoy a significant advantage in reducing the post-shock oscillations.
The remainder of this paper is organized as follows. In Section 2, we briefly review the preliminaries to understand the finite volume method and the procedures of the WENO-JS [13], WENO-M [11] and other versions of mapped WENO schemes. Section 3 presents a general method to introduce the OP mapping for improving the existing mapped WENO schemes. Some numerical results are provided in Section 4 to illustrate the advantages of the proposed WENO schemes. Finally, concluding remarks are given in Section 5 to close this paper.

Brief review of the WENO schemes
For simplicity of presentation but without loss of generality, we denote our discussion to the following onedimensional linear hyperbolic conservation equation with the initial condition u(x, 0) = u 0 (x). We only confine our attention to the uniform meshes in this paper and the WENO method with non-uniform meshes can refer to [28,12]. Throughout this paper, we assume that the given domain [x l , x r ] is discretized into the set of uniform cells I j := [x j−1/2 , x j+1/2 ], j = 1, · · · , N with the cell size ∆x = x r −x l N . The associated cell centers and cell boundaries are denoted by x j = x l + ( j − 1/2)∆x and x j±1/2 = x j ±∆x/2, respectively. The notationū(x j , t) = 1 ∆x x j+1/2 x j−1/2 u(ξ, t)dξ indicates the cell average of I j . The one-dimensional linear hyperbolic conservation equation in Eq. (2) can be approximated by a system of ordinary differential equations, yielding the semi-discrete finite volume form: whereū j (t) is the numerical approximation to the cell averageū(x j , t), and the numerical fluxf j±1/2 is a replacement of the physical flux function f (u) at the cell boundaries x j±1/2 and it is defined byf j±1/2 =f (u − j±1/2 , u + j±1/2 ). The notations u ± j±1/2 refer to the limits of u and their values of u ± j±1/2 can be obtained by the technique of reconstruction, like the WENO reconstruction procedures narrated later. In this paper, we choose the global Lax-Friedrichs flux where α = max u | f ′ (u)| is a constant and the maximum is taken over the whole range of u.

The WENO-JS reconstruction
Firstly, we review the process of the classic fifth-order WENO-JS reconstruction [13]. For brevity, we describe only the reconstruction procedure of the left-biased u − j+1/2 , and the right-biased one u + j+1/2 can trivially be computed by mirror symmetry with respect to the location x j+1/2 of u − j+1/2 . We drop the subscript "-" below just for simplicity of notation.
To construct the values of u j+1/2 from known cell average values u j , a 5-point global stencil S 5 = {I j−2 , I j−1 , I j , I j+1 , I j+2 } is used in the fifth-order WENO-JS scheme. It is subdivided into three 3-point substencils S s = {I j+s−2 , I j+s−1 , I j+s } with s = 0, 1, 2. It is known that the third-order approximations of u(x j+1/2 , t) associated with these substencils are explicitly given by Then the u j+1/2 of global stencil S 5 is computed by a weighted average of those third-order approximations of substencils, taking the form The nonlinear weights ω s in the classic WENO-JS scheme are defined as where d 0 , d 1 , d 2 are called the ideal weights of ω s since they generate the central upstream fifth-order scheme for the global stencil S 5 . It is known that d 0 = 0.1, d 1 = 0.6, d 2 = 0.3 and in smooth regions we can get 2 s=0 d s u s j+1/2 = u(x j+1/2 , t) + O(∆x 5 ). ǫ is a small positive number introduced to prevent the denominator from becoming zero. The parameters β s are the smoothness indicators for the third-order approximations u s j+1/2 and their explicit formulas can be obtained from [13], taking the form In general, the fifth-order WENO-JS scheme is able to recover the optimal convergence rate of accuracy in smooth regions. However, when at or near critical points where the first derivative vanishes but the third derivative does not simultaneously, it loses accuracy and its convergence rate of accuracy decreases to third-order or even less. We refer to [11] for more details.

The mapped WENO reconstructions
To address the issue of the WENO-JS scheme mentioned above, Henrick et al. [11] made a systematic truncation error analysis of Eq.(3) in its corresponding finite difference version by using the Taylor series expansions of the Eq. (4), and hence they derived the necessary and sufficient conditions on the weights for the fifth-order WENO scheme to achieve the formal fifth-order of convergence at smooth regions of the solution, taking the form where the superscripts "+" and "-" on ω s correspond to their use in either u s j+1/2 and u s j−1/2 stencils respectively. Since the first equation in Eq.(7) always holds due to the normalization, a simpler sufficient condition for the fifth-order convergence is given as [1] ω The conditions Eq.(7) or Eq.(8) may not hold in the case of smooth extrema or at critical points when the fifth-order WENO-JS scheme is used. An innovative idea of fixing this deficiency, originally proposed by Henrick in [11], is to design a mapping function to make ω s approximating the ideal weights d s at critical points to the required third order O(∆x 3 ). The first mapping function devised by Henrick et al. in [11] is given as We can verify that g M s (ω) meets the conditions in Eq.(8) as it is a non-decreasing monotone function on [0, 1] with finite slopes and satisfies the following properties. Lemma 1. The mapping function g M s (ω) defined by Eq.(9) satisfies: Following Henrick's idea, a great many improved mapping functions were successfully proposed [4,3,15,29,16,17,18]. To clarify our major concern and provide convenience to readers but for brevity in the description, we only state some mapping functions in the following context, and we refer to references for properties similar to Lemma 1 and more details of these mapping functions.
where c 1 , c 2 are constants with specified parameters k and d s , taking the following forms

WENO-PPMn [15]
g PPMn 5 and for n = 5, where where And m = 2, n = 6 is recommended in [29], then where with In Eq. (18), k ∈ N + , A > 0, δ > 0 with δ → 0, ǫ A is a very small positive number to prevent the denominator from becoming zero, and m s ∈ α s k+1 , M with M being a finite positive constant real number and α s a positive constant that only depends on s in the fifth-order WENO-MAIM1 scheme. In Eq. (19), the positive parameter B is a scale transformation factor introduced to adjust the shape of the mapping function and it is set to be 1 in WENO-MAIM1 while to be other values in the following WENO-ACM schemes.
WENO-ACM [17] g ACM , and k s ∈ 0, d s CFS s . By using the mapping function g X s (ω) , where the superscript "X" corresponds to "M", "PM6", or "IM", etc., the nonlinear weights of the associated WENO-X scheme are defined as where ω JS s are calculated by Eq.(6). In references, it has been analyzed and proved in detail that the WENO-X schemes can retain the optimal order of accuracy in smooth regions even at or near critical points.

3.
A general method to introduce order-preserving mapping for mapped WENO schemes

Definition 2.
(OP/non-OP point) Let S 2r−1 denote the (2r − 1)-point global stencil centered around x j . Assume that S 2r−1 is subdivided into r-point substencils {S 0 , · · · , S r−1 } and ω s are the nonlinear weights corresponding to the substencils S s with s = 0, · · · , r − 1, which are used as the independent variables by the mapping function. Suppose that g X s (ω), s = 0, · · · , r − 1 is the mapping function of the mapped WENO-X scheme, then we say that a non-OP mapping process occurs at x j , if ∃m, n ∈ {0, · · · , r − 1}, s.t.
And we say x j is a non-OP point. Otherwise, we say x j is an OP point.

Definition 3. (OP-Mapped WENO)
The family of the mapped WENO schemes with OP mappings is collectively referred to as OP-Mapped WENO in our study.

A general formula for the existing mapping functions
We rewrite the mapping function of the WENO-X scheme, that is, g X s (ω), s = 0, 1, · · · , r − 1, to be a general formula, given as g X ω; m P , P s,1 , · · · , P s, where m P is the number of the parameters related with s indicating the substencil, and P s,1 , · · · , P s,m P are these parameters. Clearly, we have m P = 0 for the WENO-JS scheme and m P ≥ 1 for other mapped WENO schemes. In Table 1, taking 9 different WENO schemes as examples, we have presented their parameters of m P and P s,1 , · · · , P s,m P . Let n X denote the order of the specified critical point, namely ω = d s , of the mapping function of the WENO-X scheme, that is, To simplify the description of Theorem 2 below, we present n X of the WENO-X scheme in the sixth column of Table 1. Table 1, the mapping function g X s (ω), s = 0, 1, · · · , r − 1 is monotonically increasing over [0, 1].

Lemma 2. For the WENO-X scheme shown in
Proof. See the corresponding references given in the last column of Table 1. Table 1. The parameters m P and P s,1 , · · · , P s,m P for the WENO-JS scheme and some existing mapped WENO schemes whose mapping functions are non-OP.

No.
Scheme, WENO-X m P P s,1 , · · · , P s,m P Parameters n X Ref

The new mapping functions
Firstly, we give the minDist function by the following definition.

Definition 4. (minDist function) Define the minDist function as follows
where d s , s = 0, · · · , r − 1 is the optimal weight, ω is the nonlinear weight being the independent variable of the mapping function, and the function IndexOf(·) returns a set of the subscripts of "·", that is, if Let D = d 0 , d 1 , · · · , d r−1 be an array of all the ideal weights of the (2r − 1)th-order WENO schemes. We build a new array by sorting the elements of D in ascending order, that is, D = d 0 , d 1 , · · · , d r−1 . In other words, the arrays D and D have the same elements with different arrangements, and the elements of D satisfy Definition 5. Let G = g X 0 (ω), g X 1 (ω), · · · , g X r−1 (ω) be an array of all the mapping functions of the (2r − 1)thorder mapped WENO-X scheme. We define a new array by sorting the elements of G in a new order, that is, G = g X 0 (ω), g X 1 (ω), · · · , g X r−1 (ω) , where g X s (ω) is the mapping function associated with d s . Proof.
(3) As the proof of the case of i = r −1 is very similar to that of the case i = 0, we do not state it here for simplicity.
(1) We can directly get properties C1 and C2 from Lemma 2.
(2) As a > b, according to Eq.(27)(35), we know that the interval Ω a must be on the right side of the interval Ω b , while ω α ∈ Ω a , ω β ∈ Ω b is given, then we get ω α > ω β . Trivially, according to Definition 5, or by intuitively observing the curves of the mapping function g X s (ω) as shown in Fig. 1 By employing the minDist function, we build a general method to introduce the OP criterion into the existing mappings which are non-OP. The general method is stated in Algorithm 1.

Algorithm 1:
A general method to construct OP mappings.
input : s, index indicating the substencil S s and s = 0, 1, · · · , r − 1 d s , optimal weights ω JS s , nonlinear weights computed by the WENO-JS scheme m P , the number of the parameters related with s P s, j , parameters related with s and j = 1, · · · , m P output: g MOP−X s (ω JS s ), s = 0, 1, · · · , r − 1 , the new set of mapping functions that is OP 1], and the set of 1 , · · · , P s,m P . 21 end Theorem 1. The set of mapping functions g MOP−X s (ω JS s ), s = 0, 1, · · · , r − 1 obtained through Algorithm 1 is OP.
Proof. Let ω JS m , ω JS n ∈ [0, 1] and ∀m, n ∈ {0, 1, · · · , r − 1}. According to Algorithm 1 and without loss of generality, we can assume that ω JS m ∈ Ω k * m , ω JS n ∈ Ω k * n , and then we get It is easy to verify that n (ω JS n ). Therefore, according to Lemma 4, we can finish the proof trivially.
We now define the modified weights which are OP as follows where g MOP−X s (ω JS s ) is obtained from Algorithm 1. The associated scheme will be referred to as MOP-WENO-X. The mapping functions of the WENO-X schemes presented in Table 1 and those of the corresponding MOP-WENO-X schemes are shown in Fig. 1. We can find that, for the mapping functions of the MOP-WENO-X schemes: (1) the monotonicity over the whole domain (0, 1) is maintained; (2) the differentiability is reduced and limited to the neighborhood of the optimal weights d s ; (3) the OP property is obtained. We summarize these properties as follows.
The mapping function g MOP−X s (ω) obtained from Althorithm 1 satisfies the following properties: where n X is given in Table 1;

Convergence properties
According to Theorem 2, we get the convergence properties for the (2r − 1)th-order MOP-WENO-X schemes as given in Theorem 3. The proof is almost identical to that of the corresponding WENO-X schemes in the references presented in Table 1.
The requirements for the (2r − 1)th-order MOP-WENO-X schemes to achieve the optimal order of accuracy are identical to that of the corresponding (2r − 1)th-order WENO-X schemes.
For the integrity of this paper and the benefit of the reader, we concisely express the following Corollaries of Theorem 3.
Proof. The proof is almost identical to that of Lemma 6 in [16].
Corollary 2. When n cp = 1, the (2r − 1)th-order MOP-WENO-IM(k, A) schemes can achieve the optimal order of accuracy if the mapping function g MOP−IM s (ω) is applied to the original weights in the (2r − 1)th-order WENO-JS schemes with requirement of k ≥ 2 (except for the case r = 2).
Proof. The proof is almost identical to that of Theorem 2 in [4].  Table 1) and MOP-WENO-X. Proof. The proof is almost identical to that of Proposition 1 in [3]. Proof. The proof is almost identical to that of Theorem 3 in [29].
Corollary 5. Let ⌈x⌉ be a ceiling function of x, for n cp < r − 1, the (2r − 1)th-order MOP-WENO-MAIM1 schemes can achieve the optimal order of accuracy if the mapping function (g MOP−MAIM1 ) s (ω) is applied to the original weights in the (2r − 1)th-order WENO-JS schemes with requirement of k ≥ k MAIM , where Proof. The proof is almost identical to that of Theorem 2 in [16]. Proof. The proof is almost identical to that of Theorem 2 in [17].
Proof. The proof is almost identical to that of Theorem 2 in [18].

Numerical results
In this section, we compare the numerical performances of the MOP-WENO-X schemes with the corresponding existing mapped WENO-X schemes shown in Table 1, as well as the classic WENO-JS scheme. As the performances of the WENO-ACM scheme and the MOP-WENO-ACM scheme are almost identical to those of the MIP-WENO-ACMk scheme and the MOP-WENO-ACMk scheme respectively, we do not present the solutions of the WENO-ACM scheme and the MOP-WENO-ACM scheme below for simplicity. Typical one-dimensional linear advection equation and two-dimensional Euler equations, with different initial conditions, are used to test the considered schemes. The presentation of these numerical tests in this section starts with the accuracy test of one-dimensional linear advection equation with four different initial conditions, followed by the long output time simulations of it with two different initial conditions including discontinuities, and finishes with two-dimensional simulations on the shock-vortex interaction and the 2D Riemann problem. In all calculations below, ǫ is taken to be 10 −40 for all schemes following the recommendations in [11,4].
In the following numerical tests, the ODEs resulting from the semi-discretized PDEs are marched in time using the following explicit, third-order, strong stability preserving (SSP) Runge-Kutta method [26,5,6] U at time level t n = n∆t, and ∆t is the time step satisfying some proper CFL condition. The spatial operator L is defined as in Eq.(3), and the WENO reconstructions will be applied to obtain it.

Accuracy test
In this subsection, we solve the following one-dimensional linear advection equation with different initial conditions to test the accuracy of the considered WENO schemes. In all accuracy tests, the L 1 , L 2 , L ∞ norms of the error are given as where h = ∆x is the uniform spatial step size, (u h ) j is the numerical solution and u exact j is the exact solution.
It is trivial to verify that although the initial condition in Eq.(38) has two first-order critical points, their first and third derivatives vanish simultaneously. The CFL number is taken to be (∆x) 2/3 to prevent the convergence rates of error from being influenced by time advancement. The calculation is run until a time of t = 2.0.
In Table 2, we show the L 1 , L 2 , L ∞ errors and corresponding convergence orders of various considered WENO schemes. The results of the three rows are L 1 -, L 2 -and L ∞ -norm errors and convergence orders (in brackets) in turn (similarly hereinafter). Unsurprisingly, the MOP-WENO-X schemes and the corresponding WENO-X schemes provide more accurate results than the WENO-JS scheme do in general. Naturally and as expected, all the considered schemes have gained the fifth-order convergence rate of accuracy. It can be found that the results of the MOP-WENO-X schemes are identical to those of the corresponding WENO-X schemes for all grid numbers except N = 10. As discussed in [18], the cause of the accuracy loss for the computing cases of all MOP-WENO-X schemes with N = 10 is that the mapping functions of the MOP-WENO-X schemes have narrower optimal weight intervals (standing for the intervals about ω = d s over which the mapping process attempts to use the corresponding optimal weights, see [16,17]) than the corresponding WENO-X schemes.
Example 2. We calculate Eq.(37) with the periodic boundary condition using the following initial condition [11] u(x, 0) = sin πx − sin(πx) π , This particular initial condition has two first-order critical points, which both have a non-vanishing third derivative. Again, the CFL number is set to be (∆x) 2/3 and the calculation is run until a time of t = 2.0. Table 3 compares the L 1 , L 2 , L ∞ errors and corresponding convergence orders obtained from the considered schemes. It is evident that the WENO-X schemes and the corresponding MOP-WENO-X schemes can achieve the optimal convergence orders. Unsurprisingly, the WENO-JS scheme gives less accurate results than the other schemes, and its L ∞ convergence order decreases by almost 2 orders leading to the noticeable drops of the L 1 and L 2 convergence orders. It is noteworthy that when the grid number is too small, like N ≤ 40, in terms of accuracy, the MOP-WENO-X schemes provide less accurate results than those of the corresponding WENO-X schemes. As mentioned in Example 1, the cause of this kind of accuracy loss is that the mapping functions of the MOP-WENO-X schemes have narrower optimal weight intervals than the corresponding WENO-X schemes, and this issue can surely be addressed by increasing the grid number. Therefore, as expected, the MOP-WENO-X schemes show equally accurate numerical solutions like those of the corresponding WENO-X schemes when the grid number N ≥ 80.
Example 3. We calculate Eq.(37) using the following initial condition [3]  u(x, 0) = sin 9 (πx), with the periodic boundary condition. It is trivial to verify that this initial condition has high-order critical points. We also set the CFL number to be (∆x) 2/3 . Table 4 shows the L 1 , L 2 , L ∞ errors of the considered WENO schemes at several output times with a uniform mesh size of ∆x = 1/200. In order to measure the dissipations of the schemes, as the authors did in [18], we give the corresponding increased errors (in percentage) compared to the MIP-WENO-ACMk scheme which gives solutions with very low dissipations. Taking the L 1 -norm error as an example, let L * 1 (t) and L Y 1 (t) the L 1 -norm errors of the MIP-WENO-ACMk scheme and the compared scheme, the increased errors at output time t is computed by Table 4, we can observe that: (1) the WENO-JS scheme shows the largest increased errors for no matter short or long output times; (2) at short output times, like t ≤ 100, the solutions computed by the WENO-M scheme are closer to those of the MIP-WENO-ACMk scheme, leading to smaller increased errors, than the corresponding MOP-WENO-M scheme; (3) however, when the output time is larger, like t ≥ 200, the solutions computed by the MOP-WENO-M scheme, whose increased errors do not get larger but evidently decrease, are closer to those of the MIP-WENO-ACMk scheme than the corresponding WENO-M scheme whose errors increase dramatically leading to significantly larger increased errors; (4) although the errors of the MOP-WENO-X schemes except the MOP-WENO-M scheme are not as small as those of the corresponding WENO-X schemes, these errors can maintain a considerable level leading to acceptable increased errors that are much lower than those of the WENO-JS and WENO-M schemes.
Actually, as mentioned in Example 1 and Example 2, the cause of the slight accuracy loss discussed above is that the mapping function of the MOP-WENO-X scheme has narrower optimal weight intervals than the corresponding Table 3. Convergence properties of considered schemes solving u t + u x = 0 with initial condition u(x, 0) = sin(πx − sin(πx)/π). WENO-X schemes, and one can easily overcome this drawback by increasing the grid number. To demonstrate this, we calculate this problem using the same schemes at the same output times with a larger grid number of N = 800. The results are shown in Table 5, and we can see that: (1) the errors of the MOP-WENO-X schemes get closer to those of the MIP-WENO-ACMk scheme when the grid number increases from N = 200 to N = 800, resulting in the significant decrease of the increased errors, and in different words, the errors of the MOP-WENO-X schemes and the MIP-WENO-ACMk scheme are so close that one can ignore their differences; (2) although the errors of the WENO-JS and WENO-M schemes get smaller when the grid number increases from N = 200 to N = 800, their increased errors become very very large; (3) naturally, the increased errors of the MOP-WENO-X schemes are extremely smaller than those of the WENO-JS and WENO-M schemes. Actually, it is an important advantage of the MOP-WENO-X schemes that can maintain comparably high resolution for long output times. In the next subsection we will make a further discussion for this. In Fig. 2 and Fig. 3, we plot the solutions computed by various schemes at output time t = 1000 with the grid number of N = 200 and N = 800, respectively. For N = 200, Fig. 2 shows that: (1) the MOP-WENO-M scheme provides the result with far higher resolution than the corresponding WENO-M scheme which gives result with slightly better resolution than the lowest one computed by the WENO-JS scheme; (2) the result of the MOP-WENO-MAIM1 scheme is very close to that of its corresponding WENO-MAIM1 scheme; (3) the results of the other MOP-WENO-X schemes show far better resolutions than the WENO-M and WENO-JS schemes, although they give results with very slightly lower resolutions than their corresponding WENO-X schemes because of the narrower optimal weight intervals. Actually, we can amend this minor issue by using a larger grid number. Consequently, for N = 800, it can be seen from Fig. 3 that: (1) all the MOP-WENO-X schemes produce results very close to those of their corresponding mapped WENO-X schemes with extremely high resolutions except the case of X = M; (2) the MOP-WENO-M scheme also produces result with very high resolution, while the resolutions of the results from the WENO-M and WENO-JS schemes are much lower. Table 4. Performance of various considered schemes solving u t + u x = 0 with u(x, 0) = sin 9 (πx), ∆x = 1/200. Example 4. We calculate Eq.(37) using the following initial condition [13] u( where 2 , 0 , and the constants are z = −0.7,δ = 0.005, β = log 2 36δ 2 , a = 0.5 and α = 10. The periodic boundary condition is used and the CFL number is taken to be 0.1. For brevity in the presentation, we call this Linear Problem SLP as it is presented by Shu et al. in [13]. It is known that this problem consists of a Gaussian, a square wave, a sharp triangle and a semi-ellipse. In Table 6, we present the L 1 , L 2 , L ∞ errors and the corresponding convergence rates of accuracy with t = 2, 2000. For the case of t = 2, it can be seen that: (1) the L 1 and L 2 orders of all considered schemes are approximately 1.0 Table 5. Performance of various considered schemes solving u t + u x = 0 with u(x, 0) = sin 9 (πx), ∆x = 1/800. and about 0.35 to 0.5, respectively; (2) negative values of the L ∞ orders of all considered schemes are generated; (3) in terms of accuracy, the MOP-WENO-X schemes produce less accurate results than the corresponding WENO-X schemes. For the case of t = 2000, it can be seen that: (1) the L 1 , L 2 orders of the WENO-JS and WENO-M schemes decrease to very small values and even become negative; (2) however, the L 1 and L 2 orders of all the MOP-WENO-X schemes, as well as the associated mapped WENO-X schemes without WENO-M, are clearly larger than 1.0 and around 0.5 to 0.9 respectively; (3) the L ∞ orders of all WENO-X schemes are very small and some of then even become negative (e.g., the WENO-JS, WENO-PPM5 and MIP-WENO-ACMk schemes), while those of the MOP-WENO-X schemes are all positive although they are also very small; (4) in terms of accuracy, on the whole, the MOP-WENO-X schemes produce accurate and comparable results as the corresponding WENO-X schemes except the WENO-M scheme. However, if we take a closer look, we can find that the resolution of the result computed by the WENO-M scheme is significantly lower than that of the MOP-WENO-M scheme, and the other mapped WENO-X schemes generate spurious oscillations but the corresponding MOP-WENO-X schemes do not. Detailed tests will be conducted and the solutions will be presented carefully to demonstrate this in the following subsection.

1D linear advection problems with long output times
The objective of this subsection is to demonstrate the advantage of the MOP-WENO-X schemes on long output time simulations that can obtain high resolution and meanwhile do not generate spurious oscillations.
The one-dimensional linear advection problem Eq.(37) is solved with the periodic boundary condition by taking the following two initial conditions. Case 1. (SLP) The initial condition is given by Eq.(41).
Case 1 and Case 2 were carefully simulated in [18]. Case 1 is called SLP as mentioned earlier in this paper. Case 2 consists of several constant states separated by sharp discontinuities at x = ±0.8, ±0.6, ±0.4, ±0.2 and it was called BiCWP for brevity in the presentation as the profile of the exact solution for this Problem looks like the Breach in City Wall.
In Figs. 4, 6 and Figs. 8, 10, we show the comparison of considered schemes for SLP and BiCWP respectively, by taking t = 2000 and N = 800. It can be seen that: (1) all the MOP-WENO-X schemes produce results with considerable resolutions which are significantly higher than those of the WENO-JS and WENO-M schemes, and what's more, they all do not generate spurious oscillations, while most of their corresponding WENO-X schemes do, when solving both SLP and BiCWP; (2) it should be reminded that the WENO-IM(2, 0.1) scheme appears not to generate spurious oscillations and it gives better resolution than the MOP-WENO-IM(2, 0.1) scheme in most of the region when solving SLP on present computing condition, however, from Fig. 4(b), one can observe that the MOP-WENO-IM(2, 0.1) scheme gives a better resolution of the Gaussian than the WENO-IM(2, 0.1) scheme, and if taking a closer look, one can see that the WENO-IM(2, 0.1) scheme generates a very slight spurious oscillation near x = −0.435 as shown in Fig. 4(c); (3) it is very evident as shown in Fig. 8 that, when solving BiCWP, the WENO-IM(2, 0.1) scheme generates the spurious oscillations.
In Figs. 5, 7 and Figs. 9, 11, we show the comparison of considered schemes for SLP and BiCWP respectively, by taking t = 200 and N = 3200. From these solutions computed with larger grid numbers and a reduced but still long output time, it can be seen that: (1) firstly, the WENO-IM(2, 0.1) scheme generates spurious oscillations but the MOP-WENO-IM(2, 0.1) scheme does not while provides an improved resolution when solving SLP; (2) although the resolutions of the results computed by the WENO-JS and WENO-M schemes are significantly improved for both SLP and BiCWP, the MOP-WENO-X schemes still evidently provide much better resolutions; (3) the spurious oscillations generated by the WENO-X schemes appear to be more evident and more intense as the grid number increases, while the corresponding MOP-WENO-X schemes can still avoid spurious oscillations but obtain higher resolutions, when solving both SLP and BiCWP.
For the further interpretation, without loss of generality, in Fig. 12 In summary, it could be indicated that the general method to introduce the OP mapping can help to gain the advantage of achieving high resolutions and in the meantime preventing spurious oscillations when solving problems 20 with discontinuities for long output times. And this is the most important point we want to report in this paper.

Euler system in two dimension
In this subsection, we focus on the numerical simulations of the shock-vortex interaction problem [2,20,22] and the 2D Riemann problem [24,23,14]. They are governed by the two-dimensional Euler system of gas dynamics, taking the following strong conservation form of mass, momentum and energy where ρ, u, v, p and E are the density, components of velocity in the x and y coordinate directions, pressure and total energy, respectively. The following equation of state for an ideal polytropic gas is used to close the two-dimensional Euler system Eq.(43) where γ is the ratio of specific heat, and we set γ = 1.4 in this paper. In the computations below, the CFL number is taken to be 0.5. All the considered WENO schemes are applied dimension-by-dimension to solve the two-dimensional Euler system and the local characteristic decomposition [13] is used. In [30], Zhang et al. investigated two commonly used classes of finite volume WENO schemes in two-dimensional Cartesian meshes, and we employ the one denoted as class A in this subsection.
We calculate this problem using all the considered mapped WENO-X schemes in Table 1  Example 6. (2D Riemann problem) It is very favorable to test the high-resolution numerical methods [14,15,21] using the series of 2D Riemann problems [24,23]. In [14], Lax   Similarly, although we calculate this problem using all the considered mapped WENO-X schemes in Table 1 and their corresponding MOP-WENO-X schemes, we only present the solutions of the WENO-M, WENO-PM6, WENO-RM260 and MIP-WENO-ACMk schemes and their corresponding MOP-WENO-X schemes here for the sake of brevity. We have shown the numerical results of density obtained by using these schemes in Figs Again, we believe that this should be an advantage of the mapped WENO schemes whose mapping functions are OP.

Conclusions
The concept of OP-Mapped WENO schemes standing for the family of the mapped WENO schemes with orderpreserving (OP) mappings, as well as a general way to build one group of this kind of schemes, has been proposed in this paper. Specifically, we extend the OP mapping introduced in [18] to various existing mapped WENO schemes in references by providing a general formula of their mapping functions. A systematic analysis has been made to prove that the improved mapped WENO scheme based on the existing mapped WENO-X scheme, denoted as MOP-WENO-X, generates numerical solutions with the same convergence rates of accuracy in smooth regions as the corresponding WENO-X scheme. Furthermore, numerical experiments are run to show that the MOP-WENO-X schemes have the same advantage as the mapped WENO scheme proposed in [18] in calculating the one-dimensional linear advection problems including discontinuities with long output times. The mapping functions of the MOP-WENO-X schemes are OP and hence able to attain high resolutions and avoid spurious oscillations meanwhile. Moreover, numerical results with the 2D Euler system problems were presented to show that the MOP-WENO-X schemes perform well in simulating the two-dimensional steady problems with strong shock waves to capture the main flow structures and remove or significantly reduce the post-shock oscillations.