A New Smoothness Indicator of Adaptive Order Weighted Essentially Non-Oscillatory Scheme for Hyperbolic Conservation Laws

: Adaptive order weighted essentially non-oscillatory scheme (WENO-AO(5,3)) has increased the computational cost and complexity of the classic ﬁfth-order WENO scheme by introducing a complicated smoothness indicator for ﬁfth-order linear reconstruction. This smoothness indicator is based on convex combination of three third-order linear reconstructions and ﬁfth-order linear reconstruction. Therefore, this paper proposes a new simple smoothness indicator for ﬁfth-order linear reconstruction. The devised smoothness indicator linearly combines the existing smoothness indicators of third-order linear reconstructions, which reduces the complexity of that of WENO-AO(5,3) scheme. Then WENO-AO(5,3) scheme is modiﬁed to WENO-O scheme with new and simple formulation. Numerical experiments in 1-D and 2-D were run to demonstrate the accuracy and efﬁcacy of the proposed scheme in which WENO-O scheme was compared with original WENO-AO(5,3) scheme along with WENO-AO-N, WENO-Z, and WENO-JS schemes. The results reveal that the proposed WENO-O scheme is not only comparable to the original scheme in terms of accuracy and efﬁcacy but also decreases its computational cost and complexity.


Introduction
It is well known that capturing shock waves in compressible flows remains a challenging issue to numerical schemes in computational fluid dynamics due to the coexistence of smooth and discontinuous structure of different scales. Thus, though the initial conditions of hyperbolic conservation laws are smooth, discontinuities might be developed in the solution. As a result, Taylor expansion-based numerical schemes fail to estimate hyperbolic conservation law solutions. Therefore, alternative schemes are proposed to realize the approximate solution without introducing spurious oscillations based on adaptively selecting the smoothest stencil among several candidates that are Essentially Non-Oscillatory (ENO) schemes. In these schemes, based on finite volume framework, Harten et al. [1] devised the concept of smoothness indicator to measure the smoothness of each interpolation polynomial and then select the smoothest stencil over different candidate stencils. Thereafter, based on finite difference discretization, Shu & Osher [2] introduced a more efficient variant of ENO schemes by directly using upwinding to the fluxes on basis of dimension-by-dimension.
In addition, a new class of ENO scheme was devised first by Liu et al. [3] and then enhanced by Jiang and Shu [4] by using more efficient smoothness indicator to reach the optimal order accuracy in smooth region. These schemes employing all candidate substencils of ENO scheme by convex combination in a non-linear manner instead of choosing the smoothest one; then for each sub-stencil, a weight is assigned according to its local coefficients as introduced in WENO-AO-N scheme. We abbreviate the new scheme as WENO-O scheme.

Finite Difference WENO Schemes
For the sake of simplicity, we adopted 1-D scalar hyperbolic conservation law where u is the conservative variables and f (u) is the flux vector. Equation (1) can be a system or a single equation. The semi-discrete finite difference scheme of (1) is and u i (t) denotes the numerical approximation to the point value of u(x i , t), R(u) i is given by: wheref i±1/2 represent numerical fluxes at the cell boundaries that are calculated using a numerical scheme (see Figure 1). Here we consider uniform mesh to divide the spatial domain, i.e., h = x i+1 − x i , x i+1/2 = x i + h/2 and the cells are denoted by The property of conservation is obtained by implicitly defining the function v(x) through: where v(x) approximates the numerical fluxf i±1/2 with a high order of accuracy, i.e., To ensure the numerical stability, the flux f (u) is split into two parts f + (u) and f − (u) with d f + (u) du ≥ 0 and d f − (u) du ≤ 0, using the global Lax-Friedrichs flux splitting method aŝ where α is taken as max u f (u) andf i+1/2 =f − i+1/2 +f + i+1/2 . Sincef − i+1/2 is symmetric to the positive part with respect to x i+1/2 , from here onwards, we will only describe hoŵ f + i+1/2 is constructed and the "+ sign in the superscript will be dropped for convenience.

Fifth-Order WENO-JS Scheme
Basically, WENO schemes [4] are designed to approximate the spatial derivative for solving the hyperbolic conservation laws in essentially non-oscillatory manner. In the fifth-order WENO scheme, the five-points stencil (7) of sub-stencils flux functions is used to reconstruct the flux functionf i+1/2 at the cell I i interface x i+1/2 . The main difference between WENO and ENO schemes is that WENO schemes use the convex combination of sub-stencils flux functions instead of using smoothest sub-stencil, which is written aŝ The WENO non-linear weights ω k (8) are designed to approach very small value when the corresponding sub-stencil contains a discontinuity and approach the optimal value for smooth solution.
where α k (k = 0, 1, 2) are referred to as the un-normalized weights, ε is small number used to prevent division by zero, p = 2. To measure the smoothness of a solution over a particular sub-stencil, the smoothness indicators β k are employed which are given by: In addition, d k represent the ideal/optimal weights as they create the upstream central fifth-order scheme for the five-point stencil in smooth regions aŝ The flux functionf i+1/2 of the cell interface is reconstructed using sub-stencils' polynomialsf k (x i+1/2 ) as in (11). The values of ideal weights are given by where g j (j = i − 2, i − 1, i, i + 1, i + 2) are known cell averages, and the smoothness indicators are, explicitly, given by

Fifth-Order WENO-Z Scheme
Later, Henrick et al. [5] found that for fifth-order convergence, the non-linear weights (8) do not satisfy the necessary and sufficient conditions thus to enhance the accuracy of these weights they introduced a mapping function. However, Borges et al. [6] devised different approach to improve the non-linear weights of original WENO scheme by using the absolute difference between β 2 and β 0 then they proposed a new set of WENO weights as Therefore, WENO-Z scheme includes higher-order information achieved from the global order smoothness indicator, τ 5 , in the formation of the non-linear weights as where p = 2, and the global smoothness indicator for the fifth-order WENO-Z scheme is given by

The Adaptive Order WENO Scheme
Dinshaw et al. [9] presented a new class of WENO schemes, known as WENO-AO, using adaptive order property as a non-linear hybridization between a rock-stable r = 3 CWENO scheme and a higher-order centered stencil. Therefore, they proposed WENO-AO(5,3) scheme (16) that is at best fifth-order accurate by virtue of its centered stencil with five zones, and at worst third-order accurate by virtue of being non-linearly hybridized with an r = 3 CWENO scheme.
And the non-linear weights are determined as follows: with where h and l stand for high (fifth) and low (third) order. In addition, at inflection points, WENO-AO(5,3) scheme [9] uses the τ parameter to avoid loss of order.

A New and Simple Smoothness Indicator
Therefore, and inspiring by [24,25], we propose new and simple smoothness indicator denoted asβ h (22), which can be used to replace β h in (20) andβ h in (21). The new smoothness indicator is constructed based on an idea of linearly combining the existing smoothness indicators of third-order linear reconstructions; this will lead to reduce the complexity of that of WENO-AO and WENO-AON schemes. The new smoothness indicator is given byβ where d 0 , d 1 and d 2 are the optimal weights. It can be seen that the proposed smoothness indicator (22) is simple in the explicit form since it uses existing sub-stencils smoothness indicators (β 0 , β 1 and β 2 ) unlike β h of WENO-AO(5,3) scheme. Moreover, it is without non-linear convex combination and complex coefficients asβ h of WENO-AO-N scheme. By usingβ h , a new WENO-AO(5,3) reconstruction is achieved, which is called WENO-O scheme. Since the five-points stencil polynomial could be constructed using optimal weights and the polynomials of sub-stencil aŝ Equation (16) might be written aŝ Thus, a new non-linear weight can be derived as with simplifying (25) we get where δ i = (d i −w l i )/w h , By adding non-linear weights (17) into (26) we have or, explicitly: It is easy to see that ∑ 2 j=0 ω O j = 1 and ∑ 2 j=0 δ j = 1 with δ 0 = 95/900, δ 1 = 510/900, and δ 2 = 295/900. The un-normalized weights (α h , α 0 , α 1 , and α 2 ) are determined using (19). As a result, the fifth-order WENO-O scheme flux function is given bŷ Moreover, a comparison of the proposed WENO-O scheme with original adaptive order scheme WENO-AO and modified adaptive order scheme WENO-AON is presented in Tables 1 and 2. These tables demonstrated that the proposed WENO-O scheme has reduced the complexity of both WENO-AO and WENO-AON schemes in which the flux function is in simple formulation and smoothness indicator is just one-step calculated equation. Table 1. Comparison of flux function of the adaptive order schemes (WENO-AO and WENO-AON) and proposed WENO-O scheme.

Scheme
Flux Function

Analysis of the Smoothness Indicators
The observation of the Taylor series of the smoothness indicators β h ,β h , andβ h reveals no difference in the order of accuracy, as will be discussed in this analysis. By using the Taylor series of f (x) at x i+1/2 for β 0 , β 1 and β 2 (12) we have For simplicity, P denotes the f (x i+1/2 ). Moreover, by using the Taylor series for the smoothness indicators of WENO-AO, WENO-AON and WENO-O schemes, β h (20),β h (21), andβ h (22), respectively, at x i+1/2 , we obtain As can be seen from this analysis, the three smoothness indicators, β h ,β h , andβ h , have the same leading and second terms which reveals that the new smoothness indicator β h gives comparable values in smooth region of solution as the original one β h .

Time Integration
Using method-of-lines approach, Equation (1) is estimated by the third-order three stages TVD Runge-Kutta method of [26] that is given by with ∆t and n are the time step and time level, respectively.

Numerical Experiments
To demonstrate the accuracy and convergence rate of the devised WENO-O scheme, eight test cases are studied and shown. To perform a fair comparison, the original WENO-AO, WENO-AON, WENO-Z, and WENO-JS schemes are adopted to assess the performance of the new WENO-O scheme. To calculate the errors in L 1 −, L ∞ − norms the exact solution u exact i is compared with the numerical solutionū i with, using where N is the cell number.
The numerical computations were run to a final time of t = 10 with CFL = (∆x) 2/3 . This case was used to test the convergence rate of the new scheme WENO-O because, as time evolves, the solution is always smooth. For all considered fifth-order WENO schemes, L 1 -norm errors, L ∞ -norm errors, computational time and the accuracy are presented in Table 3 with different mesh numbers. In this table, it is shown that all the methods reach their design accuracies. On comparing the new WENO scheme (WEON-O) with both adaptive order schemes (WENO-AO and WENO-AON), for the same mesh, the new scheme exhibits less computational time which is good to see for decreasing the adaptive order schemes' computational cost. Furthermore, the proposed WENO-O scheme presents similar efficiency as both adaptive order schemes, as can be seen from Figures 2 and 3.

Example (2)-Non-Linear Burgers Equation in One Dimension
On the one-dimensional periodic domain, we use the non-linear Burgers equation with initial condition We run numerical simulations to a final time of t = 1 with CFL = 0.1 on different uniform mesh numbers for all considered WENO schemes as described in Example 1 (Section 6.1.1). The accuracy results along with L 1 -norm errors and computational time are given in Table 4. We see that our WENO-O scheme is somewhat better than WENO-AO scheme in terms of computational time. Moreover, the efficiency of the proposed method is also better than adaptive order scheme WENO-AO as can be seen from Figures 4 and 5. However, the new WENO-O scheme gives more accurate solution, on the same mesh, than WENO-AO scheme (see Figure 5). Moreover, the values of smoothness indicatorsβ h , β h , andβ h of WENO-O, WENO-AO, and WENO-AON schemes, respectively, are displayed in Figure 6 in which the value of smoothness indicator of WENO-O scheme showed closer results to that of WENO-AON and WENO-AO schemes.

T h e s m o o t h n e s s i n d i c a t o r
x

Example (3)-The Scalar Advection Test Problem
The 1-D periodic domain x ∈ (−1, 1) is considered to solve the scalar advection equation where and the constants are δ = 0.005, α = 10, z = −0.7, a = 0.5 and β = log2/36δ 2 . This test case is conducted to assess the ability of the devised WENO-O scheme to resolve rich constructions of the solution that contains a smooth but narrow combination of a square wave, Gaussian, half ellipse, and sharp triangle wave. The simulations are performed at t = 10 and CFL = 0.1 with different meshes. The accuracy of considered fifth-order WENO schemes is presented in Table 5, in which L 1 -norm errors, computational time and convergence rates display that the solution of the new scheme is comparable to the adaptive order WENO schemes. Moreover, WENO-O scheme reduces the computational cost of the adaptive order schemes; this also can be observed from Figure 7. In addition, the results are plotted in Figure 8 for 1600 mesh points and the close view of some locations is presented in Figure 9. It could be observed that the devised scheme exhibited better solution than other schemes and then higher efficiency. Moreover, convergence rates of all schemes dropped to first order from fifth order, due to discontinuities in the solution.

One-Dimensional Euler Equations
The three well-known cases governed by the 1-D Euler Equation (38) are examined in Examples 4-6.
∂U ∂t here U and F represent conservative vector and convective flux in x direction, respectively, which are U = (ρ, ρu, E) T , where E, p, u and ρ denote the total energy, pressure, velocity component in x direction and the density, respectively. The pressure is calculated by where γ denotes the specific heat ratio (γ = 1.4 for all considered cases).

Example (4)-The Lax Test Problem
To assess the capability of the devised WENO-O scheme to capture relatively strong shock, the Lax problem is used [27]. The initial conditions are  Table 6 and Figure 11; however, our scheme reveals less L 1 -error norm with the same grids than WENO-AO scheme. Moreover, it gives less computational time than that of the original WENO-AON and WENO-AO schemes. Moreover, the results exhibited that the new WENO-O scheme provides correct solutions and good resolution to the shock wave as accurate as adaptive order WENO schemes (WENO-AON and WENO-AO schemes) and the contact discontinuity can be resolved sharply as illustrated in Figures 12 and 13 .

Example (5)-The Sod Test Problem
The 1-D Euler Equation (38) were solved on x ∈ [−5, 5] using This Riemann problem [28] that contains contact discontinuity, shock, and rarefaction waves is simulated at t = 2 and CFL = 0.1 with different uniform meshes. The results of the L 1 error norms, convergence rates and computational time are summarized in Table 7 and Figure 14. A similar observation as in Example 4 (Section 6.2.1) can be concluded from this example. The efficiency of the devised scheme for predicting the solution of Sod's problem is illustrated in Figures 15 and 16. Thus, the devised WENO-O scheme predicts the contact discontinuity, shock, and rarefaction waves as accurate as adaptive order and WENO-Z schemes and better than WENO-JS scheme.
The shock entropy wave interaction problem of Shu-Osher [29] is also adopted to test the new WENO-O scheme. The numerical solutions are run with CFL number 0.1 to final time of t = 1.8. Since the exact solution is unavailable, the solution of WENO-AO scheme that obtained over mesh size of 6400 is adopted as the reference solution. From Figures 17 and 18, it shows the similar conclusion as in Example 4 (Section 6.2.1) for both accuracy and efficiency of proposed WENO-O scheme.

Two-Dimensional Euler Equations
To further test the proposed WENO-O scheme, two test cases for two-dimensional coordinates (2-D) are simulated; two of which are two-dimensional Riemann problems along with double-Mach reflection of strong shock. Therefore, the 2-D Euler Equation (42) are used in this work.

∂U ∂t
here U is the conservative vector, F and G are the convective fluxes in x and y directions, respectively, which are U = (ρ, ρu, ρv, E) T , F(U) = (ρu, ρu 2 + p, ρuv, u(E + p)) T , G(U) = (ρv, ρuv, ρv 2 + p, v(E + p)) T , with E, p, u, v, p and ρ denote the total energy, pressure, velocity components in x and y directions and density, respectively. The pressure is calculated by and γ denotes the specific heat ratio (γ = 1.4 for all considered cases).

Example (7)-Double-Mach Reflection of Strong Shock
Woodward and Colella [30] have proposed the well-known 2-D Double-Mach shock reflection test case that had been taken extensively to assess high resolution schemes. In this case, a strong vertical shock moves horizontally into a wedge that is inclined with some angle. The considered WENO schemes were used to simulate a 2-D computational domain of [0, 4] × [0, 1] whereas the reflective wall lies on the bottom of the computational domain for 1/6 ≤ x ≤ 4. In the beginning, a right-moving Mach 10 shock is positioned at x = 1/6, y = 0 and creates an angle 60 • with the x-axis. More details can be found elsewhere [30]. We consider the 2-D Euler Equation (42)

CPU Time Comparison
As shown in Section 3 (Tables 1 and 2), the devised WENO-O scheme has decreased the complexity of adaptive order schemes (WENO-AO and WENO-AON). In this section, a comparison of CPU time of the devised WEON-O scheme and adaptive order schemes (WENO-AO and WENO-AON) is presented for six test cases (see Figure 22). In this figure, the CPU time of original WENO-AO scheme is used as a reference to other schemes (WENO-O and WENO-AON) because both schemes are modified version of WENO-AO scheme. The first modification of original WENO-AO scheme, WENO-AON scheme, showed less CPU time for all six test cases compared to WENO-AO scheme. The devised WENO-O scheme illustrated less CPU time than both original WENO-AO and WENO-AON schemes. One can conclude that the devised WENO-O scheme has decreased the computational cost of both WENO-AO and WENO-AON schemes.

Conclusions
It was shown that in previously published literature, adaptive order weighted essentially non-oscillatory WENO-AO(5,3) has increased the computational cost and complexity by introducing a complicated smoothness indicator for fifth-order linear reconstruction. Therefore, in this work we have proposed a simple smoothness indicator for fifth-order linear reconstruction. The devised smoothness indicator linearly combines the existing smoothness indicators of third-order linear reconstructions which reduces the complexity of that of WENO-AO(5,3) scheme. Then, the adaptive order weighted essentially non-oscillatory scheme uses the devised smoothness indicator is known as WENO-O scheme. We have demonstrated through eight numerical experiments that the WENO-O scheme has decreased the complexity and computational cost of adaptive order schemes (WENO-AO(5,3) and WENO-AON).