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The calculation of the weight of each sub-stencil is very important for a weighted essentially non-oscillatory (WENO) scheme to obtain high-order accuracy in smooth regions and keep the essentially non-oscillatory property near discontinuities. The weighting function introduced in the WENO-Z scheme provides a straightforward method to analyze the a...
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Citations
... The distribution of density along = 5 for various TENO schemes. condition of this case at the square region [0, 10] × [0, 10] is given as[44] ...
... In contrast, the scheme will decrease the correction of nonlinear weights and is more dissipative with q = 2. Hence, Borges et al. [9] suggested the power q taken to be 1 to maintain the low dissipation property. Recently, some studies [18][19][20][21][22][23] reconstructed a series of higher-order (even up to eighth-order) reference smoothness indicators to recover the optimal convergence for the low-dissipation WENO-Z scheme (means q = 1). Although numerical results confirmed that these modified schemes with higher-order smoothness indicators can achieve the desired accuracy in smooth regions including critical points, most of these works either used larger stencils or introduced user-tunable parameters, which led to the increased computation time or limited the application. ...
... However, an increase in q resulted in more numerical dissipation of WENO-Z. Therefore, the power parameter q = 1 is suggested by Borges et al. and used by other scholars [18][19][20][21][22][23][24][25]. ...
... Some works [20][21][22][23] demonstrated that constructing more higher-order smoothness indicators was useful in recovering the desired order for the low-dissipation WENO-Z scheme (means q = 1). Unlike these studies, we reconstructed a novel fifth-order global smoothness indicator with lower dissipation for the WENO-Z scheme to overcome its inherent limitations, such as its failure to balance the low-dissipation property and fifthorder convergence. ...
The fifth-order WENO-Z scheme proposed by Borges et al., using a linear combination of low-order smoothness indicators, is designed to provide a low numerical dissipation to solve hyperbolic conservation laws, while the power q in the framework of WENO-Z plays a key role in its performance. In this paper, a novel global smoothness indicator with fifth-order accuracy, which is based on several lower-order smoothness indicators on two-point sub-stencils, is presented, and a new lower-dissipation WENO-Z scheme (WENO-NZ) is developed. The spectral properties of the WENO-NZ scheme are studied through the ADR method and show that this new scheme can exhibit better spectral results than WENO-Z no matter what the power value is. Accuracy tests confirm that the accuracy of WENO-Z with q = 1 would degrade to the fourth order at first-order critical points, while WENO-NZ can recover the optimal fifth-order convergence. Furthermore, numerical experiments with one- and two-dimensional benchmark problems demonstrate that the proposed WENO-NZ scheme can efficiently decrease the numerical dissipation and has a higher resolution compared to the WENO-Z scheme.
... And Jiang & Shu (1996) constructed third-and fifthorder-accurate weight functions for the FD WENO scheme, which has been widely used for studies of hydrodynamic processes. Since then, several WENO schemes with modified weight functions have been suggested, such as WENO-M (Henrick et al. 2005), WENO-Z (Borges et al. 2008), WENO-CU (Hu et al. 2010), WENO-NS (Ha et al. 2013), WENO-ZA (Liu et al. 2018), and WENO-NIP (Li & Zhong 2022). And Jiang & Wu (1999) built an MHD code based on the WENO scheme by Jiang & Shu (1996). ...
Due to the prevalence of magnetic fields in astrophysical environments, magnetohydrodynamic (MHD) simulation has become a basic tool for studying astrophysical fluid dynamics. To further advance the precision of MHD simulations, we have developed a new simulation code that solves ideal adiabatic or isothermal MHD equations with high-order accuracy. The code is based on the finite-difference weighted essentially nonoscillatory (WENO) scheme and the strong stability-preserving Runge–Kutta (SSPRK) method. Most of all, the code implements a newly developed, high-order constrained transport (CT) algorithm for the divergence-free constraint of magnetic fields, completing its high-order competence. In this paper, we present the version in Cartesian coordinates, which includes a fifth-order WENO and a fourth-order five-stage SSPRK, along with extensive tests. With the new CT algorithm, fifth-order accuracy is achieved in convergence tests involving the damping of MHD waves in 3D space. And substantially improved results are obtained in magnetic loop advection and magnetic reconnection tests, indicating a reduction in numerical diffusivity. In addition, the reliability and robustness of the code, along with its high accuracy, are demonstrated through several tests involving shocks and complex flows. Furthermore, tests of turbulent flows reveal the advantages of high-order accuracy and show that the adiabatic and isothermal codes have similar accuracy. With its high-order accuracy, our new code would provide a valuable tool for studying a wide range of astrophysical phenomena that involve MHD processes.
... Clearly, even at points in which the first derivative equal zero of the smooth solution, these indicators can retrain the seventh order. However, when solving problems involving shock waves, numerical results showed that these higher-order global smoothness indicators may cause oscillations [9]. Conversely, employing the parameter ε prevents the denominator from being equal to zero. ...
... Jiang and Shu [6] calculate a classical formula for . the accessible format of for the 7th-order WENO scheme that may be proved as (9) Note: The right value is acquired through symmetry. ...
This study presents a modified seventh-order weighted essentially non-oscillatory (WENO) finite difference scheme based on the numerical perturbation method established in [1]. The perturbed candidate polynomials of the seventh-order WENO scheme are evolved using a perturbational polynomial of the grid spacing, which modifies the polynomial approximation used for the classical WENO7-Z reconstruction on each candidate stencil. Furthermore, it is found that the new weighted scheme constructed with the new perturbed polynomials candidate has necessary and sufficient conditions for seventh-order convergence that are one order lower than those used by Henrick for the classic WENO scheme with seventh-order convergence, as presented in [2]. As a result, even at critical locations, the new seventh-order WENO scheme, which uses the perturbed polynomials and the same weights as the WENO7-Z scheme as demonstrated in [3], is able to satisfy the necessary and sufficient condition for seventh-order convergence.
The new WENO7-P scheme reduces numerical dissipation in WENO schemes. Numerical examples verify the new scheme's accuracy, low dissipation, and robustness.
... They developed a class of structurally simple fifth-order WENO-Z schemes. In addition, Castro et al. (2011), Don and Borges (2013), Liu et al. (2018), Wang et al. (2018), Peng et al. (2019), Baeza et al. (2019aBaeza et al. ( , b, 2020, , Semplice et al. (2016), Dumbser et al. (2017), Cravero et al. (2018), Rathan et al. (2020) and Huang and Chen (2021) have successively developed various high-order WENO-Z type schemes based on Borges'. ...
To improve the shock-capturing capability of the third-order WENO scheme and enhance its computational efficiency, in this paper, we design a new WENO scheme independent of the local smoothing factor, WENO-SIF. The weight function of the WENO-SIF scheme is the segmentation function of the sub-stencil, which is guaranteed to achieve the desired accuracy at higher order critical points. WENO-SIF does not need to compute the smoothing factor during the computation, which effectively reduces the computational consumption. The present WENO-SIF is compared with WENO-JS and other WENO schemes for numerical experiments at one- and two-dimensional benchmark problems with a suitable choice of \(\lambda =0.13\). The results demonstrate that the WENO scheme can further improve the resolution of WENO-JS, achieve optimal accuracy at high-order critical points, and significantly reduce the computational consumption.
... And Jiang & Shu (1996) constructed third and fifth-order accurate weight functions for the FD WENO scheme, which has been widely used for studies of hydrodynamic processes. Since then, several WENO schemes with modified weight functions have been suggested, such as WENO-M (Henrick et al. 2005), WENO-Z (Borges et al. 2008), WENO-CU (Hu et al. 2010), WENO-NS (Ha et al. 2013), WENO-ZA (Liu et al. 2018), and WENO-NIP (Li & Zhong 2022). And Jiang & Wu (1999) built an MHD code based on the WENO scheme by Jiang & Shu (1996). ...
Due to the prevalence of magnetic fields in astrophysical environments, magnetohydrodynamic (MHD) simulation has become a basic tool for studying astrophysical fluid dynamics. To further advance the precision of MHD simulations, we have developed a new simulation code that solves ideal adiabatic or isothermal MHD equations with high-order accuracy. The code is based on the finite-difference weighted essentially non-oscillatory (WENO) scheme and the strong stability-preserving Runge-Kutta (SSPRK) method. Most of all, the code implements a newly developed, high-order constrained transport (CT) algorithm for the divergence-free constraint of magnetic fields, completing its high-order competence. In this paper, we present the version in Cartesian coordinates, which includes a fifth-order WENO and a fourth-order five-stage SSPRK, along with extensive tests. With the new CT algorithm, fifth-order accuracy is achieved in convergence tests involving the damping of MHD waves in three-dimensional space. And substantially improved results are obtained in magnetic loop advection and magnetic reconnection tests, indicating a reduction in numerical diffusivity. In addition, the reliability and robustness of the code, along with its high accuracy, are demonstrated through several tests involving shocks and complex flows. Furthermore, tests of turbulent flows reveal the advantages of high-order accuracy, and show the adiabatic and isothermal codes have similar accuracy. With its high-order accuracy, our new code would provide a valuable tool for studying a wide range of astrophysical phenomena that involve MHD processes.
... Compared to the mapped WENO scheme, the WENO-Z scheme does not need to map the weight function, which can effectively reduce the time of calculation. After that, researchers developed various global smoothing factors to improve the WENO-Z scheme, such as, Castro et al. (2011), Don and Borges (2013), Fan et al. (2014), Liu et al. (2018), Peng et al. (2019), . ...
... The solution until the final time t = 3.2 s using uniform meshes with 400 × 400 grids are shown in Fig. 16, where 11 equidistant values ranging from 0.1 to 0.2. One can see that all four schemes have good symmetry and are rich in vortex structures at the interfaces, while the vortex structures of VWENO-JS and VWENO-Z are significantly clear than those of WENO-JS and WENO-Z.4.3.5 Double-Mach reflection problemThe initial conditions for this problem ofLiu et al. (2018),Peng et al. (2019),,Shi et al. (2003),Tang and Li (2020) in the computational region [0, 4] × [0, 1] are given as(ρ, u, v, p) = (8, 57.1597, −33.0012, 563.544), ...
In this paper, a new type of weighted essentially non-oscillatory scheme with variable index (VWENO) is obtained. The index can adaptively adjust with the solution, to ensure that the VWENO scheme uses optimal weights in smooth regions, while non-linear weights are used in less smooth regions. Theoretical and numerical results show that the variable index can make the result of VWENO achieve the optimal weights in the smooth regions without amplifying the weight of less smooth sub-stencils containing discontinuities. Theoretical and numerical calculation experiments show that the new scheme’s shock capture capability and the resolution of complex process structures are significantly better than WENO-JS and WENO-Z.
... They developed a new fifth-order WENO-Z (WENO-D) by adding a function that can correct the convergence accuracy to the weight function used with the WENO-Z method. In addition, other scholars have also developed WENO-Z-type schemes by constructing various global smoothness indicators that satisfy different requirements [7,28,35,42,44]. ...
In this paper, we developed a novel high-resolution fifth-order WENO-JS-type scheme that can achieve the optimal accuracy order at or near the arbitrary order critical points. To enhance the accuracy of the weight function of WENO-JS, we normalized the local smoothness factor to obatain a series of normalized variables. By introducing a mapping function that can increase the accuracy of the normalization variables, we developed a new normalized WENO-JS-type scheme called WENO-NSI. The WENO-NSI scheme can improve the shock capture ability by selecting the apposite parameter λ in the mapping function. Numerical experiments of one- and two-dimensional gas dynamic Euler equations for the present WENO-NSI scheme are compared with those of the WENO-JS/Z/M/D/IM and TENO-Z schemes. The results show that the present scheme can achieve theoretical accuracy at or near the arbitrary order critical points, and it has a better shock capturing ability than other schemes.
... This sample test is usually employed to verify nonoscillatory type and shock-capturing schemes to evaluate their non-oscillatory property and numerical dissipation (Deng et al., 2019;Zhang et al., 2021). This test case was also considered in previous studies by Su and Kim (2018), Huang and Chen (2018), and Liu et al. (2018), Christlieb et al., (2019), Deng et al., (2019), and Ha and Lee (2020). However, they generally considered the problem for short-term simulations, i.e., T ≤ 1000 periods. ...
Engquist et al. (1989) proposed non-linear TVD filters. When combined with a traditional higher-order finite difference scheme, these filters can simulate shock or high concentration gradient problems with no spurious oscillations. Among the filter proposed is the filter algorithm 2.2. However, this algorithm flattens extrema that are not the results of overshooting and consequency the scheme reduces to a low order of accuracy locally around smooth extrema. Modification of the TVD filter algorithm 2.2 has been proposed in this paper to overcome this problem. Several conservative finite difference schemes are considered for testing the TVD filter. Non-conservative schemes consisting of 4th and 6th -order Runge-Kutta method are also evaluated. The modified filter has been tested to simulate seven test cases, including a pure advection of scalar profiles, a pure advection with variable velocity, two inviscid burger equations, an advection-diffusion equation with variable velocity and dispersion, advection of three solid bodies in rotating fluid around a square of a side length of 2, and a two-dimensional advection-diffusion equation. The numerical experiments showed that applying the modified TVD filter, combined with the higher-order non-TVD finite difference schemes for solving the advection equation, can produce accurate solutions with no oscillations and no clipping effect extrema.
... They termed the scheme as WENO-Z. Based on the improvement of local and global smoothness indicators, many WENO schemes have been developed in the literature [14,[24][25][26][27][28][29][30][31][32][33]. Some of these schemes are WENO-CU6 [14], WENO-NS [28], WENO-P [30], and WENO-Z+ [31]. ...
In this work a hybrid alternative mapped weighted essentially non-oscillatory (HAW-M) scheme has been developed for solving non-linear hyperbolic conservation laws. In the HAW-M scheme, we have combined a new improved alternative mapped WENO (IAWENO-M) reconstruction procedure with the sixth-order central scheme (or linear scheme). The hybridization is achieved using a weighted switch based on improved non-l,oinear weights. The third-order TVD Runge–Kutta method has been used for the time advancement of the solution. The computations have been performed for various one, two, and three-dimensional test cases. Numerical results are compared with the exact solutions and results obtained with other high-resolution schemes. The proposed hybrid scheme provides sufficient numerical dissipation for capturing strong shock waves and low numerical dissipation to resolve fine structures. Further, it shows better conservation of kinetic energy for the 3D Taylor–Green vortex case.
