Fifth-order Weighted Essentially Non-oscillatory Research Articles

Deep smoothness weighted essentially non-oscillatory method for two-dimensional hyperbolic conservation laws: A deep learning approach for learning smoothness indicators

Deep smoothness weighted essentially non-oscillatory method for two-dimensional hyperbolic conservation laws: A deep learning approach for learning smoothness indicators

Hybrid conservative central/WENO finite difference scheme for two-dimensional detonation problems

In the present work, a hybrid finite difference scheme is proposed and tested for the study of two-dimensional detonation problems. The hybrid scheme consists of a fifth-order weighted essentially nonoscillatory (WENO) method for discontinuous regions and a fourth or sixth-order robust conservative central finite difference scheme for the remaining domain. An arbitrary shock sensor is used to identify the discontinuities. The proposed shock sensor is based on the absolute value of the density gradient with an arbitrary threshold computed at each time step using a fast image segmentation technique. The sensor is complemented with a Ducros sensor to avoid the selection of vortices as discontinuities. The hybrid scheme is tested for several benchmark examples, including a two-dimensional transverse detonation wave, detonation wave diffraction, and a multiple obstacles test case. The obtained results show that the proposed sensor detects discontinuities adequately and the hybrid schemes give the same results as the fifth-order WENO in faster computational time.

HOW-MHD: A High-order WENO-based Magnetohydrodynamic Code with a High-order Constrained Transport Algorithm for Astrophysical Applications

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.

A unified framework for non-linear reconstruction schemes in a compact stencil. Part 1: Beyond second order

Discretization of the convection term in a three-cell compact stencil has been widely used in Computational Fluid Dynamics (CFD) of engineering because the second-order interpolation scheme is considered to be more robust and algorithmically simple than very high-order schemes. However, according to Godnunov's theorem, it has never been trivial to design an accurate and oscillation-free discretization scheme in a compact stencil. Over decades, various kinds of schemes have been proposed to increase accuracy beyond second-order without producing non-physical numerical oscillations. Representative schemes are polynomial-based non-linear interpolations such as TVD (Total Variation Diminishing), NVD (Normalized Variable Diagram) and third-order WENO (Weighted Essentially Non-oscillatory); and non-polynomial-based interpolations including RBF (Radial Basis Functions), LDLR (Local Double Logarithmic Reconstruction) and Tangent of Hyperbola for INterface Capturing (THINC). Within the semi-discretization finite volume method, existing reconstruction schemes suffer from different issues to some extent. For example, existing TVD schemes tend to cause smearing, clipping and squaring effects. Most WENO schemes and inconsistent RBF schemes are not scale-invariant. Especially, the improved WENO-Z scheme is neither scale-invariant nor mesh-size-independent. Computationally expensive non-polynomial-based interpolations usually have singularity and are dependent on tunable parameters. Thus, this work reviews reconstruction schemes by reformulating or calculating the reconstructed boundary value in the normalised-variable space where the characteristics of reconstruction schemes can be read directly. Through the accuracy analysis and the induction of existing schemes, this work proposes a Unified Normalised-variable Diagram (UND) where the following properties of a reconstruction scheme can be given diagrammatically: 1) the second-order and the third-order accuracy condition, 2) the TVD and CBC (Convection Boundedness Criterion) constrain, 3) the ENO condition, 4) the High Resolution (HR) property, 5) and the numerical dissipation and anti-dissipation error.The proposed UND framework is firstly demonstrated by the improvement of the existing schemes with a shape-preserving TVD limiter and a scale-invariant WENO scheme. Then, by defining the normalised reconstruction operator, a brand-new scheme named ROUND (Reconstruction Operator on Unified Normalised-variable Diagram) is proposed. The ROUND scheme is devised by directly configuring the normalised reconstruction operator of favourable properties on UND. This work presents a dissipative ROUND scheme in which normalised reconstruction operators are fully in the dissipative region of UND, and a low-dissipative ROUND scheme where rigorously adjusted anti-dissipation errors are introduced. Different formulations of ROUND schemes are presented to show the flexibility of designing schemes within the UND framework. The proposed dissipative and low-dissipative ROUND schemes are evaluated via benchmark tests which show the superior performance of ROUND on accuracy and resolution. Especially, in some cases, the low-dissipative ROUND schemes obtain comparable and even better resolution than classic fifth-order WENO. The significance of this work is further demonstrated by the preliminary results of applying ROUND on the FDM-IBM (Finite Difference Method and Immersed Boundary Method) framework.

A hybrid WENO scheme for steady-state simulations of Euler equations

For strong shock waves in solutions of steady-state Euler equations, the high-order shock capturing schemes usually suffer from the difficulty of convergence of residue close to machine zero. Several new weighted essentially non-oscillatory (WENO) type schemes have recently been designed to overcome this long-standing issue. In this paper, a new hybrid strategy is proposed for the fifth-order WENO scheme to simulate steady-state solutions of Euler equations. Compared with the existing WENO schemes, the hybrid WENO scheme performs better steady-state convergence property with less dissipative and dispersive errors. Meanwhile, the essentially oscillation-free feature is kept. In the hybrid strategy, the stencil is distinguished into smooth, non-smooth, or transition regions, which is realized by a simple smoothness detector based on the smoothness indicators in the original WENO method. The linear reconstruction and the specific WENO reconstruction are applied to the smooth and non-smooth regions, respectively. In the transition region, the mixture of the linear and WENO reconstructions is adopted by a smooth transitive interpolation, which plays a vital role in the steady-state convergence for the hybrid scheme. Numerical comparisons and spectral analysis are presented to demonstrate the robust performance of the new hybrid scheme for steady-state Euler equations.

Numerical investigation on noise reduction of rotor blade-vortex interaction using blade surface jet blowing

The effects of blade surface jet blowing on the reduction of rotor blade-vortex interaction (BVI) noise are investigated using the compressible Reynolds-averaged Navier-Stokes equations and the Ffowcs Williams-Hawkings equations. A detailed analysis of the accuracy of the Weighted Essentially Non-Oscillatory (WENO) scheme and Monotone Upstream-Centered Scheme for Conservation Laws for simulating the flow field and predicting the BVI noise is performed using a two-bladed scaled AH-1 helicopter main rotor undergoing BVI as the baseline case. Additionally, a grid convergence study is conducted. The BVI noise is accurately predicted using the fifth-order WENO scheme and high-resolution grid system (the prediction errors of the peak sound pressure are less than 4%). Surface boundary conditions are introduced to model the effect of jet blowing on the blade surface on the flow control of the baseline case. Blade surface blowing is analyzed near the trailing edge and the leading edge near the rotor tip. The results show that blowing near the blade trailing edge is effective for BVI mitigation by altering the tip vortex strength and increasing the blade-vortex miss distance. The sound pressure level is reduced by 2.6 dB, and the peak amplitude is attenuated by more than 30%, with a 9.8% loss in rotor thrust. Therefore, blowing near the trailing edge may be considered a promising approach for active BVI noise control.

Two-dimensional numerical simulation on gas-solid two-phase flow induced by combustion gas flow in a chamber based on a fifth-order WENO scheme

In order to explore the flow characteristics and propagation law of combustion gas from the central ignition tube in the initial stage of internal ballistic, a visual experimental platform was designed to carry out ignition experiments with the substituted particle bed in the chamber. A high-speed camera system was used to capture the gas flow and flame propagation in the chamber, and a dynamic data acquisition and analysis system with pressure sensors was applied to record the pressure data at characteristic positions in the chamber. A two-dimensional, axisymmetrical, two-phase flow model of internal ballistics was constructed to simulate the flow process of the gas in the substituted particle bed based on a weighted essentially non-oscillatory (WENO) scheme, and the time term was determined by the third-order TVD Runge-Kuta method. The calculated results are in good agreement with the visual experimental results, and the global pressure average error is 5.35%. It indicates that the numerical simulation can accurately describe the gas flow characteristics and present the development process of the gas from the ignition tube in the substituted particle bed. The radial effect of the chamber pressure is obvious, and the gas moves rapidly along the radial direction, and the substituted particle basically does not move in the initial stage of ignition. Moreover, with the gradual propagation of the gas in the chamber, the chamber pressure is characterized by a radial uniformity and an axial gradient distribution. Under the action of the pressure gradient, the axial velocity of the gas phase begins to dominate, and furthermore, the radial velocity decreases to zero in the bottom and the middle region of the chamber, while the solid phase velocity varies with the gas phase velocity. In addition, before the gas reaches the bottom of the right end, the inverse velocity fluctuation appears in advance due to the solid particle congestion.

A fifth-order high-resolution shock-capturing scheme based on modified weighted essentially non-oscillatory method and boundary variation diminishing framework for compressible flows and compressible two-phase flows

First, a new reconstruction strategy is proposed to improve the accuracy of the fifth-order weighted essentially non-oscillatory (WENO) scheme. It has been noted that conventional WENO schemes still suffer from excessive numerical dissipation near-critical regions. One of the reasons is that they tend to under-use all adjacent smooth substencils thus fail to realize optimal interpolation. Hence in this work, a modified WENO (MWENO) strategy is designed to restore the highest possible order interpolation when three target substencils or two target adjacent substencils are smooth. Since the new detector is formulated under the original smoothness indicators, no obvious complexity and cost are added to the simulation. This idea has been successfully implemented into two classical fifth-order WENO schemes, which improve the accuracy near the critical region but without destroying essentially non-oscillatory properties. Second, the tangent of hyperbola for interface capturing (THINC) scheme is introduced as another reconstruction candidate to better represent the discontinuity. Finally, the MWENO and THINC schemes are implemented with the boundary variation diminishing algorithm to further minimize the numerical dissipation across discontinuities. Numerical verifications show that the proposed scheme accurately captures both smooth and discontinuous flow structures simultaneously with high-resolution quality. Meanwhile, the presented scheme effectively reduces numerical dissipation error and suppresses spurious numerical oscillation in the presence of strong shock or discontinuity for compressible flows and compressible two-phase flows.

A new fifth-order alternative finite difference multi-resolution WENO scheme for solving compressible flow

A new fifth-order alternative finite difference multi-resolution weighted essentially non-oscillatory (WENO) scheme is designed to solve the hyperbolic conservation laws in this paper. With the application of the original finite difference multi-resolution WENO schemes (Zhu and Shu, 2018; Zhu and Shu, 2020), a series of unequal-sized central spatial stencils are adopted to perform the WENO procedures, but the difference is that the methodology adopted in this paper is directly based on the point values of the solution rather than on the flux values. The proposed new multi-resolution WENO scheme can inherit many advantages of the original high order schemes, such as the arbitrary choice of the linear weights, the maintenance of the essentially non-oscillatory property in the vicinity of strong discontinuities, the smaller L1 and L∞ truncation errors than that of the same order classical WENOJS schemes (Balsara et al., 2016; Jiang and Shu, 1996) in smooth regions, and better convergence properties of the residue when solving some steady-state problems. Compared with the original multi-resolution WENO schemes, the presented WENO scheme also has some advantages. Firstly, an arbitrary monotone flux can be used in this framework, while the original method is only suitable for smooth flux splitting technique. Secondly, it has smaller L1 and L∞ truncation errors than that of the original same order multi-resolution WENO scheme. And finally, it avoids some time-consuming intermediate processes in the original multi-resolution WENO reconstruction procedures, thus resulting in less calculation time under the same conditions. Some inviscid and viscous numerical examples are provided to verify the superior performance of the new fifth-order alternative finite difference multi-resolution WENO scheme.

Comparison Study of the Fifth-Order WCNS and WENO Scheme In the Truncation Error, Dissipation and Dispersion Terms

This paper makes a comparison study of the fifth-order weighted compact nonlinear scheme (WCNS) and weighted essentially non-oscillatory (WENO) scheme in the truncation error, dissipation and dispersion terms. Truncation error analysis on the WCNS shows that numerical discretization leads to dissipation and dispersion terms. The Fourier spectral analysis methods is used to analyze the dissipative and dispersive features of both schemes.

A fourth-order entropy condition scheme for systems of hyperbolic conservation laws

Based on the analysis of the entropy condition scheme formulation, the accuracy order comes from initial value interpolation and flux reconstruction. Following the limiters of the traditional second-order Total Variation Diminishing scheme, higher accuracy order and non-oscillatory nature are retained with a newly proposed smoothness threshold method. Then, the scheme using the solution formula method in Dong et al. [(2002). High-order discontinuities decomposition entropy condition schemes for Euler equations. CFD Journal, 10(4), 563–568] is generalized to fully-discrete fourth-order accuracy, which retains the advantages of former schemes, i.e. it is a fully-discrete, one-step scheme with no need to perform with a Runge–Kutta process in time; an entropy condition is satisfied with no need of an entropy fix with artificial numerical viscosity; and an exact solution is achieved for linear cases if CFL→1. Numerical experiments are given with a 1D scalar equation for a shock-tube problem, a blast-wave problem, and Shu–Osher problem, a 2D Riemann problem, a double Mach reflection problem and a transonic airfoil flow problem for NACA0012. All tests are compared with a fifth-order Weighted Essentially Non-Oscillatory (WENO) scheme. Numerical experiments and efficiency comparisons show that the efficiency of the new fourth-order scheme is superior to the fifth-order WENO scheme.

A low dissipation finite difference nested multi-resolution WENO scheme for Euler/Navier-Stokes equations

In this paper, a low dissipation fifth-order finite difference nested multi-resolution weighted essentially non-oscillatory (WENO) scheme is presented for solving Euler/Navier-Stokes equations in multi-dimensions on structured meshes. In the reconstruction procedures of this finite difference WENO, a series of nested unequal-sized central stencils are used to design spatial reconstruction polynomials based on the local orthogonal Legendre basis functions. So this new WENO scheme could maintain the original order of accuracy in smooth regions and keep essentially non-oscillatory property by gradually degrading from the fifth-order to third-order, and ultimately to the first-order accuracy near strong discontinuities. Compared with the classical fifth-order finite difference WENO scheme [18], the results show that this new nested multi-resolution WENO scheme could obtain smaller L1 and L∞ errors for solving smooth problems based on the same mesh level. Moreover, the presented WENO scheme also has smaller numerical dissipation without introducing any dissipation preserving technique, and can capture more subtle flow structures for solving viscous flow problems. The proposed multi-resolution WENO scheme can be easily extended to finite volume framework owing to its linear weights can be set as any positive numbers with only one requirement that their summation equals to one. Therefore, the new WENO scheme is more suitable for solving viscous problems containing both discontinuities and complex smooth structures, and can be easily implemented to arbitrarily high-order accuracy in multi-dimensions. Some benchmark inviscid extreme and viscous examples are illustrated to verify the good performance of this nested multi-resolution WENO scheme.

A new fifth-order finite difference well-balanced multi-resolution WENO scheme for solving shallow water equations

In this paper, a new fifth-order finite difference well-balanced multi-resolution weighted essentially non-oscillatory (WENO) scheme is designed to solve for one-dimensional and two-dimensional shallow water equations with or without source terms on structured meshes. We only use the information defined on a hierarchy of nested central spatial stencils without introducing any equivalent multi-resolution representation once again. For the shallow flow problems with smooth or discontinuous bed, we combine with the well-balanced procedure developed by Xing and Shu (2005) for balancing the flux gradients and the source terms, and then the new fifth-order well-balanced multi-resolution WENO scheme (Zhu and Shu, 2018) could satisfy the exact C-property for still stationary solutions, maintain the fifth-order accuracy in smooth regions, and keep essentially non-oscillatory property in non-smooth regions. Compared with the classical well-balanced WENO schemes (Xing and Shu, 2005), the new features of this finite difference well-balanced multi-resolution WENO scheme is its simplicity and hierarchical structure in obtaining higher-order accuracy and its linear weights could be arbitrarily chosen with one requirement that their summation is one. It is the first time that a series of unequal-sized hierarchical central spatial stencils are used in designing finite difference well-balanced WENO scheme for solving shallow water equations. This new well-balanced WENO scheme is simple to construct and can be easily implemented to arbitrary high-order accuracy and in higher dimensions. Some benchmark numerical examples are performed to illustrate the good performances of this new well-balanced WENO scheme which could get high-order accuracy, keep exact C-property, sustain good convergence property, and obtain sharp shock transitions in the whole computational field.

A new type of multi-resolution WENO schemes with increasingly higher order of accuracy on triangular meshes

In this paper, we continue our work in [46] and propose a new type of high-order finite volume multi-resolution weighted essentially non-oscillatory (WENO) schemes to solve hyperbolic conservation laws on triangular meshes. Although termed “multi-resolution WENO schemes”, we only use the information defined on a hierarchy of nested central spatial stencils and do not introduce any equivalent multi-resolution representation. We construct new third-order, fourth-order, and fifth-order WENO schemes using three or four unequal-sized central spatial stencils, different from the classical WENO procedure using equal-sized biased/central spatial stencils for the spatial reconstruction. The new WENO schemes could obtain the optimal order of accuracy in smooth regions, and could degrade gradually to first-order of accuracy so as to suppress spurious oscillations near strong discontinuities. This is the first time that only a series of unequal-sized hierarchical central spatial stencils are used in designing arbitrary high-order finite volume WENO schemes on triangular meshes. The main advantages of these schemes are their compactness, robustness, and their ability to maintain good convergence property for steady-state computation. The linear weights of such WENO schemes can be any positive numbers on the condition that they sum to one. Extensive numerical results are provided to illustrate the good performance of these new finite volume WENO schemes.

Numerical simulation of shock-disturbances interaction in high-speed compressible inviscid flow over a blunt nose using weighted essentially non-oscillatory scheme

In the present study, shock-disturbances interaction in high-speed compressible inviscid flow is simulated utilizing the weighted essentially non-oscillatory (WENO) scheme by employing the shock-capturing technique. For this aim, the two-dimensional Euler equations in strong conservative form are discretized by using the explicit third-order TVD Runge–Kutta scheme in time and the fifth-order WENO finite difference scheme in space. The main advantage of using the WENO scheme is its capability for properly solving the discontinuities in the domain without needing any artificial viscosity, limiter function or filter. Hence, this scheme is stable, and thus, it is suitable for simulating very high-speed flows with strong shocks. Three numerical tests are presented to demonstrate the remarkable capability of the WENO scheme and the corresponding numerical results are compared with the available analytical and numerical studies in the literature. The final and main test case is the steady and unsteady numerical simulation of the high-speed compressible inviscid flow past a two-dimensional circular cylinder. For this test case, firstly, the steady numerical solution at Mach=8.03 is provided. Then, the disturbances field is added to the upstream flow field and the unsteady numerical solution is performed for the fast and slow acoustic and also entropy waves. A sensitivity study is also performed to check the effects of the grid size and the type of flux splitting method on the solution. Both the steady and unsteady numerical solutions are compared with the existing analytical and numerical solutions and some conclusions are made.

Preconditioned WENO finite-difference lattice Boltzmann method for simulation of incompressible turbulent flows

In this work, a preconditioned high-order weighted essentially non-oscillatory (WENO) finite-difference lattice Boltzmann method (WENO-LBM) is applied to deal with the incompressible turbulent flows. Two different turbulence models namely, the Spalart–Allmaras (SA) and k−ωSST models are used and applied in the solution method for this aim. The spatial derivatives of the two-dimensional (2D) preconditioned LB equation in the generalized curvilinear coordinates are discretized by using the fifth-order WENO finite-difference scheme and an implicit–explicit Runge–Kutta scheme is adopted for the time discretization. For the convective and diffusive terms of the turbulence transport equations, the third-order WENO and second-order central finite-difference schemes are used, respectively. The preconditioning technique along with the local time-stepping method are applied to the WENO-LBM to further accelerate the solution to the converged steady-state condition, and therefore, an accurate and efficient incompressible LB solver is provided for simulating turbulent flows. The accuracy and robustness of the proposed solution method are assessed by computing two test cases: the 2D turbulent flow over a flat plate at Re=1.0×107 and the 2D turbulent flow over a NACA0012 airfoil at Re=6.0×106 and different angles of attack. The present results are compared with the available numerical and experimental results which show excellent agreement. It is demonstrated that the present solution methodology based on the WENO-LBM provides more accurate results of the incompressible turbulent flows and it requires lower number of grid points compared to the traditional (low-order accurate) LB and Navier–Stokes solvers.

An Optimized Symmetric WENO Method-Based Numerical Simulation of Intense Sound Field Generated by Underwater Plasma Sound Source

The intensive pulse sound wave can be generated by the underwater plasma sound source (UPSS) based on the discharge of the underwater high voltage. The distribution of the sound field is prominently nonlinear. In this paper, the sound field of the intensive UPSS is described by the integral two-dimensional axisymmetric unsteady Euler equations firstly. In order to solve the Euler equations numerically, an optimized fifth-order symmetric WENO (weighted essentially nonoscillatory) method based on the three templates is proposed which is called WENO-SYM3. Without increasing the number of candidate templates, a new symmetric template structure can be obtained by expanding the second template and shifting the third one backwards for one space. The method is validated through numerical examples and experiments, and the results show that WENO-SYM3 has a high distinguished accuracy; meanwhile, its nonphysical oscillations are not obvious. The experimental results are basically the same as the calculation results, and the maximum error is around 3%.

Numerical study on the convergence to steady-state solutions of a new class of finite volume WENO schemes: triangular meshes

In this paper, we continue our research on the numerical study of convergence to steady-state solutions for a new class of finite volume weighted essentially non-oscillatory (WENO) schemes in Zhu and Shu (J Comput Phys 349:80–96, 2017), from tensor product meshes to triangular meshes. For the case of triangular meshes, this new class of finite volume WENO schemes was designed for time-dependent conservation laws in Zhu and Qiu (SIAM J Sci Comput 40(2):A903–A928, 2018) for the third- and fourth-order versions. In this paper, we extend the design to a new fifth-order version in the same framework to keep the essentially non-oscillatory property near discontinuities. Similar to the case of tensor product meshes in Zhu and Shu (2017), by performing such spatial reconstruction procedures together with a TVD Runge–Kutta time discretization, these WENO schemes do not suffer from slight post-shock oscillations that are responsible for the phenomenon wherein the residues of classical WENO schemes hang at a truncation error level instead of converging to machine zero. The third-, fourth-, and fifth-order finite volume WENO schemes in this paper can suppress the slight post-shock oscillations and have their residues settling down to a tiny number close to machine zero in steady-state simulations in our extensive numerical experiments.

A sixth-order finite difference WENO scheme for Hamilton–Jacobi equations

ABSTRACTIn this paper, a sixth-order finite difference weighted essentially non-oscillatory (WENO) scheme is developed to approximate the viscosity solution of the Hamilton–Jacobi equations. This new WENO scheme has the same spatial nodes as the classical fifth-order WENO scheme proposed by Jiang and Peng [Weighted ENO schemes for Hamilton–Jacobi equations, SIAM. J. Sci. Comput. 21 (2000), pp. 2126–2143] but can be as high as sixth-order accurate in smooth region while keeping sharp discontinuous transitions with no spurious oscillations near discontinuities. Extensive numerical experiments in one- and two-dimensional cases are carried out to illustrate the capability of the proposed scheme.

A modified fifth-order WENO scheme for hyperbolic conservation laws

Recently a WENO scheme, with smoothness indicators constructed based on L1 measure is introduced by Ha et al. (2013) and the improved version of this scheme is presented by Kim et al. (2016), referred to as WENO-NS and WENO-P schemes respectively. These schemes perform better than the existing many fifth-order WENO schemes for the problems which contain discontinuities and attain fifth-order accuracy at the critical points where the first derivative vanishes but not at the points where the second derivatives are zero. This paper deals with modification of the above said methods to obtain a new fifth-order weighted essentially non-oscillatory (WENO) scheme. A new global-smoothness indicator is proposed which shows an improved behavior over the solutions of WENO-NS and WENO-P schemes and the proposed scheme attains an optimal order of approximation, even at the critical points where the first and second derivatives vanish but not the third derivative. Examples are taken in the numeric section to check the robustness and accuracy of the proposed scheme for one and two-dimensional system of Euler equations.