Novel local smoothness indicators for improving the third‐order WENO scheme

The local smoothness indicators play an important role in the performance of a weighted essentially nonoscillatory (WENO) scheme. Due to having only 2 points available on each substencil, the local smoothness indicators calculated by conventional methods make the third‐order WENO scheme too dissipative. In this paper, we propose a different method to calculate the indicators by using all the 3 points on the global stencil of the third‐order WENO scheme. The numerical results demonstrate that the WENO scheme with the new indicators has less dissipation and better resolution than the conventional third‐order WENO scheme of Jiang and Shu for both smooth and discontinuous solutions.     

Simulation of multiple shock–shock interference using implicit anti‐diffusive WENO schemes

Accurate computations of two‐dimensional turbulent hypersonic shock–shock interactions that arise when single and dual shocks impinge on the bow shock in front of a cylinder are presented. The simulation methods used are a class of lower–upper symmetric‐Gauss–Seidel implicit anti‐diffusive weighted essentially non‐oscillatory (WENO) schemes for solving the compressible Navier–Stokes equations with Spalart–Allmaras one‐equation turbulence model. A numerical flux of WENO scheme with anti‐diffusive flux correction is adopted, which consists of first‐order and high‐order fluxes and allows for a more flexible choice of first‐order dissipative methods. Experimental flow fields of type IV shock–shock interactions with single and dual incident shocks by Wieting are computed. By using the WENO scheme with anti‐diffusive flux corrections, the present solution indicates that good accuracy is maintained and contact discontinuities are sharpened markedly as compared with the original WENO schemes on the same meshes. Computed surface pressure distribution and heat transfer rate are also compared with experimental data and other computational results and good agreement is found. Copyright © 2009 John Wiley & Sons, Ltd.     

A new smoothness indicator for third‐order WENO scheme

We present a new reference smoothness indicator for third‐order weighted essentially non‐oscillatory scheme to recover its design‐order convergence at critical points. This reference smoothness indicator, which involves both the candidate and global smoothness indicators in the weighted essentially non‐oscillatory framework, is devised according to a sufficient condition on the weights for third‐order convergence. The recovery of design‐order is verified by standard tests. Meanwhile, numerical results demonstrate that the present reference smoothness indicator produces sharper representation of the discontinuity owing to the combined effects of larger weight assignment to the discontinuous stencils and convergence rate recovery. Copyright © 2015 John Wiley & Sons, Ltd.     

Third‐order WENO scheme with a new smoothness indicator

In this article, we have devised a new reference smoothness indicator for third‐order weighted essentially non‐oscillatory (WENO) scheme to achieve desired order of convergence at critical points. In the context of the weighted essentially non‐oscillatory scheme, reference smoothness indicator is constructed in such a way that it satisfies the sufficient condition on the weights for the third‐order convergence. The goal is to construct a reference smoothness indicator such that the resulted scheme have to achieve the required order of accuracy even if the first two derivatives vanish but not the third derivative. The construction of such reference smoothness indicator is not possible through a linear combination of local smoothness indicators only. We have proposed a reference smoothness indicator to be of the fourth order of accuracy on three‐point stencil that contains the linear combination of the first derivative information of the local and global stencils. The performance enhancement of the WENO scheme through this reference smoothness indicator is verified through the standard numerical experiments. Numerical results indicate that the new scheme provides better results in comparison with the earlier third‐order WENO schemes like WENO‐JS and WENO‐Z. Copyright © 2017 John Wiley & Sons, Ltd.     

High‐order ENO and WENO schemes for unstructured grids

This work describes the implementation and analysis of high‐order accurate schemes applied to high‐speed flows on unstructured grids. The class of essentially non‐oscillatory schemes (ENO), that includes weighted ENO schemes (WENO), is discussed in the paper with regard to the implementation of third‐ and fourth‐order accurate methods. The entire reconstruction process of ENO and WENO schemes is described with emphasis on the stencil selection algorithms. The stencils can be composed by control volumes with any number of edges, e.g. triangles, quadrilaterals and hybrid meshes. In the paper, ENO and WENO schemes are implemented for the solution of the dimensionless, 2‐D Euler equations in a cell centred finite volume context. High‐order flux integration is achieved using Gaussian quadratures. An approximate Riemann solver is used to evaluate the fluxes on the interfaces of the control volumes and a TVD Runge–Kutta scheme provides the time integration of the equations. Such a coupling of all these numerical tools, together with the high‐order interpolation of primitive variables provided by ENO and WENO schemes, leads to the desired order of accuracy expected in the solutions. An adaptive mesh refinement technique provides better resolution in regions with strong flowfield gradients. Results for high‐speed flow simulations are presented with the objective of assessing the implemented capability. Copyright © 2007 John Wiley & Sons, Ltd.     

An improved unstructured WENO method for compressible multi‐fluids flow

An improved high‐order accurate WENO finite volume method based on unstructured grids for compressible multi‐fluids flow is proposed in this paper. The third‐order accuracy WENO finite volume method based on triangle cell is used to discretize the governing equations. To have higher order of accuracy, the P1 polynomial is reconstructed firstly. After that, the P2 polynomial is reconstructed from the combination of the P1. The reconstructed coefficients are calculated by analytical form of inverse matrix rather than the numerical inversion. This greatly improved the efficiency and the robustness. Four examples are presented to examine this algorithm. Numerical results show that there is no spurious oscillation of velocity and pressure across the interface and high‐order accurate result can be achieved. Copyright © 2016 John Wiley & Sons, Ltd.     

A semi‐Lagrangian multi‐moment finite volume method with fourth‐order WENO projection

Numerical oscillation has been an open problem for high‐order numerical methods with increased local degrees of freedom (DOFs). Current strategies mainly follow the limiting projections derived originally for conventional finite volume methods and thus are not able to make full use of the sub‐cell information available in the local high‐order reconstructions. This paper presents a novel algorithm that introduces a nodal value‐based weighted essentially non‐oscillatory limiter for constrained interpolation profile/multi‐moment finite volume method (CIP/MM FVM) (Ii and Xiao, J. Comput. Phys., 222 (2007), 849–871) as an effort to pursue a better suited formulation to implement the limiting projection in schemes with local DOFs. The new scheme, CIP‐CSL‐WENO4 scheme, extends the CIP/MM FVM method by limiting the slope constraint in the interpolation function using the weighted essentially non‐oscillatory (WENO) reconstruction that makes use of the sub‐cell information available from the local DOFs and is built from the point values at the solution points within three neighboring cells, thus resulting a more compact WENO stencil. The proposed WENO limiter matches well the original CIP/MM FVM, which leads to a new scheme of high accuracy, algorithmic simplicity, and computational efficiency. We present the numerical results of benchmark tests for both scalar and Euler conservation laws to manifest the fourth‐order accuracy and oscillation‐suppressing property of the proposed scheme. Copyright © 2016 John Wiley & Sons, Ltd.     

Robustness of the hybrid DRP‐WENO scheme for shock flow computations

This paper describes a new variant of hybrid scheme that is constructed by a wave‐capturing scheme and a nonoscillatory scheme for flow computations in the presence of shocks. The improved fifth‐order upwind weighted essentially nonoscillatory scheme is chosen to be conjugated with the seven‐point dispersion‐relation‐preserving scheme by means of an adaptive switch function of grid‐point type. The new hybrid scheme can achieve a better resolution than the hybrid scheme which is based on the classical weighted essentially scheme. Ami Harten's multiresolution analysis algorithm is applied to density field for detecting discontinuities and setting point values of the switch function adaptively. Moreover, the tenth‐order central filter is applied in smooth part of the flow field for damping dispersion errors. This scheme can promote overall computational efficiency and yield oscillation‐free results in shock flows. The resolution properties and robustness of the new hybrid scheme are tested in both 1D and 2D linear and nonlinear cases. It performs well for computing flow problems with rich structures of weak/strong shocks and large/small vortices, such as the shock‐boundary layer interaction problem in a shock tube, which illustrates that it is very robust and accurate for direct numerical simulation of gas‐dynamics flows. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical experiments with several variant WENO schemes for the Euler equations

Numerical experiments with several variants of the original weighted essentially non‐oscillatory (WENO) schemes (J. Comput. Phys. 1996; 126 :202–228) including anti‐diffusive flux corrections, the mapped WENO scheme, and modified smoothness indicator are tested for the Euler equations. The TVD Runge–Kutta explicit time‐integrating scheme is adopted for unsteady flow computations and lower–upper symmetric‐Gauss–Seidel (LU‐SGS) implicit method is employed for the computation of steady‐state solutions. A numerical flux of the variant WENO scheme in flux limiter form is presented, which consists of first‐order and high‐order fluxes and allows for a more flexible choice of low‐order schemes. Computations of unsteady oblique shock wave diffraction over a wedge and steady transonic flows over NACA 0012 and RAE 2822 airfoils are presented to test and compare the methods. Various aspects of the variant WENO methods including contact discontinuity sharpening and steady‐state convergence rate are examined. By using the WENO scheme with anti‐diffusive flux corrections, the present solutions indicate that good convergence rate can be achieved and high‐order accuracy is maintained and contact discontinuities are sharpened markedly as compared with the original WENO schemes on the same meshes. Copyright © 2008 John Wiley & Sons, Ltd.     

An adaptive multilevel wavelet framework for scale‐selective WENO reconstruction schemes

We put forth a dynamic computing framework for scale‐selective adaptation of weighted essential nonoscillatory (WENO) schemes for the simulation of hyperbolic conservation laws exhibiting strong discontinuities. A multilevel wavelet‐based multiresolution procedure, embedded in a conservative finite volume formulation, is used for a twofold purpose. (i) a dynamic grid adaptation of the solution field for redistributing grid points optimally (in some sense) according to the underlying flow structures, and (ii) a dynamic minimization of the in built artificial dissipation of WENO schemes. Taking advantage of the structure detection properties of this multiresolution algorithm, the nonlinear weights of the conventional WENO implementation are selectively modified to ensure lower dissipation in smoother areas. This modification is implemented through a linear transition from the fifth‐order upwind stencil at the coarsest regions of the adaptive grid to a fully nonlinear fifth‐order WENO scheme at areas of high irregularity. Therefore, our computing algorithm consists of a dynamic grid adaptation strategy, a scale‐selective state reconstruction, a conservative flux calculation, and a total variation diminishing Runge‐Kutta scheme for time advancement. Results are presented for canonical examples drawn from the inviscid Burgers, shallow water, Euler, and magnetohydrodynamic equations. Our findings represent a novel direction for providing a scale‐selective dissipation process without a compromise on shock capturing behavior for conservation laws, which would be a strong contender for dynamic implicit large eddy simulation approaches.     

A new weighting method for improving the WENO‐Z scheme

The calculation of the weight of each substencil is very important for a weighted essentially nonoscillatory (WENO) scheme to obtain high‐order accuracy in smooth regions and keep the essentially nonoscillatory property near discontinuities. The weighting function introduced in the WENO‐Z scheme provides a straightforward method to analyze the accuracy order in smooth regions. In this paper, we construct a new sixth‐order global smoothness indicator (GSI‐6) and a function about GSI‐6 and the local smoothness indicators (IS_k) to calculate the weights. The analysis and numerical results show that, with the new weights, the scheme satisfies the sufficient condition for the fifth‐order convergence in smooth regions even at critical points. Meanwhile, it can also maintain low dissipation for discontinuous solutions due to relative large weights assigned to discontinuous substencils.     

A novel weighting switch function for uniformly high‐order hybrid shock‐capturing schemes

Hybrid schemes are very efficient for complex compressible flow simulation. However, for most existing hybrid schemes in literature, empirical problem‐dependent parameters are always needed to detect shock waves and hence greatly decrease the robustness and accuracy of the hybrid scheme. In this paper, based on the nonlinear weights of the weighted essentially non‐oscillatory (WENO) scheme, a novel weighting switch function is proposed. This function approaches 1 with high‐order accuracy in smooth regions and 0 near discontinuities. Then, with the new weighting switch function, a seventh‐order hybrid compact‐reconstruction WENO scheme (HCCS) is developed. The new hybrid scheme uses the same stencil as the fifth‐order WENO scheme, and it has seventh‐order accuracy in smooth regions even at critical points. Numerical tests are presented to demonstrate the accuracy and robustness of both the switch function and HCCS. Comparisons also reveal that HCCS has lower dissipation and less computational cost than the seventh‐order WENO scheme. Copyright © 2016 John Wiley & Sons, Ltd.     

GPU‐accelerated direct numerical simulations of decaying compressible turbulence employing a GKM‐based solver

Gas Kinetic Method‐based flow solvers have become popular in recent years owing to their robustness in simulating high Mach number compressible flows. We evaluate the performance of the newly developed analytical gas kinetic method (AGKM) by Xuan et al. in performing direct numerical simulation of canonical compressible turbulent flow on graphical processing unit (GPU)s. We find that for a range of turbulent Mach numbers, AGKM results shows excellent agreement with high order accurate results obtained with traditional Navier–Stokes solvers in terms of key turbulence statistics. Further, AGKM is found to be more efficient as compared with the traditional gas kinetic method for GPU implementation. We present a brief overview of the optimizations performed on NVIDIA K20 GPU and show that GPU optimizations boost the speedup up‐to 40x as compared with single core CPU computations. Hence, AGKM can be used as an efficient method for performing fast and accurate direct numerical simulations of compressible turbulent flows on simple GPU‐based workstations. Copyright © 2016 John Wiley & Sons, Ltd.     

Applications of a higher‐order bounded numerical scheme to turbulent flows

This paper applies the higher‐order bounded numerical scheme Weighted Average Coefficients Ensuring Boundedness (WACEB) to simulate two‐ and three‐dimensional turbulent flows. In the scheme, a weighted average formulation is used for interpolating the variables at cell faces and the weighted average coefficients are determined from a normalized variable formulation and total variation diminishing (TVD) constraints to ensure the boundedness of the solution. The scheme is applied to two turbulent flow problems: (1) two‐dimensional turbulent flow around a blunt plate; and (2) three‐dimensional turbulent flow inside a mildly curved U‐bend. In the present study, turbulence is evaluated by using a low‐Reynolds number version of the k–ω model. For the flow simulation, the QUICK scheme is applied to the momentum equations while either the WACEB scheme (Method 1) or the UPWIND scheme (Method 2) is used for the turbulence equations. The present study shows that the WACEB scheme has at least second‐order accuracy while ensuring boundedness of the solutions. The present numerical study for a pure convection problem shows that the ‘TVD’ slope ranges from 2 to 4. For the turbulent recirculating flow, two different mixed procedures (Method 1 and Method 2) produce a substantial difference for the mean velocities as well as for the turbulence kinetic energy. Method 1 predicts better results than Method 2 does, comparing the analytical solution and the experimental data. For the turbulent flow inside the mildly curved U‐bend, although the predictions of velocity distributions with two procedures are very close, a noticeable difference of turbulence kinetic energy is exhibited. It is noticed that the discrepancy exists between numerical results and the experimental data. The reason is the limit of the two‐equation turbulence model to such complex turbulent flows with extra strain‐rates. Copyright © 2001 John Wiley & Sons, Ltd.     

基于HLL-HLLC的高阶WENO格式及其应用研究

HLL-HLLC格式能够克服HLLC在强激波附近的激波不稳定现象,并且保持了HLLC的低耗散特性,是一种适合更大马赫数范围的近似黎曼求解器。本文从RANS方程出发,将HLL-HLLC近似黎曼求解器结合五阶WENO重构,实现了对无粘通量的高阶离散;同时,采用完全守恒形式的四阶中心差分格式处理粘性项,建立了RANS 方程的高阶数值求解格式。通过对四个经典算例,钝头体、 ONERA M6机翼、DLR F6-WB翼身组合体和DLR F6-WBNP复杂外形的数值模拟,考察了两种WENO改进格式在复杂流场中的表现,研究了高阶格式的收敛特性;给出了在复杂流动中 WENO自由参数的推荐值,以增强求解的收敛性。算例结果表明,本文构造的高阶格式鲁棒性好,能够显著改善激波位置和激波强度,捕获更丰富的流场细节,满足复杂工程应用需求。     

Performance of very‐high‐order upwind schemes for DNS of compressible wall‐turbulence

The purpose of the present paper is to evaluate very‐high‐order upwind schemes for the direct numerical simulation (DNS ) of compressible wall‐turbulence. We study upwind‐biased (UW ) and weighted essentially nonoscillatory (WENO ) schemes of increasingly higher order‐of‐accuracy (J. Comp. Phys. 2000; 160 :405–452), extended up to WENO 17 (AIAA Paper 2009‐1612, 2009). Analysis of the advection–diffusion equation, both as Δx→0 (consistency), and for fixed finite cell‐Reynolds‐number Re_Δx (grid‐resolution), indicates that the very‐high‐order upwind schemes have satisfactory resolution in terms of points‐per‐wavelength (PPW ). Computational results for compressible channel flow (Re∈[180, 230]; M?_CL ∈[0.35, 1.5]) are examined to assess the influence of the spatial order of accuracy and the computational grid‐resolution on predicted turbulence statistics, by comparison with existing compressible and incompressible DNS databases. Despite the use of baseline O(Δt²) time‐integration and O(Δx²) discretization of the viscous terms, comparative studies of various orders‐of‐accuracy for the convective terms demonstrate that very‐high‐order upwind schemes can reproduce all the DNS details obtained by pseudospectral schemes, on computational grids of only slightly higher density. Copyright © 2009 John Wiley & Sons, Ltd.     

Entropy dissipation scheme and minimums‐increase‐and‐maximums‐decrease slope limiter

In this paper, we continue to study the entropy dissipation scheme developed in former. We start with a numerical study of the scheme without the entropy dissipation term on the linear advection equation, which shows that the scheme is stable and numerical dissipation and numerical dispersion free for smooth solutions. However, the numerical results for discontinuous solutions show nonlinear instabilities near jump discontinuities. This is because the scheme enforces two related conservation properties in the computation. With this study, we design a so‐called ‘minimums‐increase‐and‐maximums‐decrease’ slope limiter in the reconstruction step of the scheme and delete the entropy dissipation in the linear fields and reduce the entropy dissipation terms in the nonlinear fields. Numerical experiments show improvements of the designed scheme compared with the results presented in former. However, the minimums‐increase‐and‐maximums‐decrease limiter is still not perfect yet, and better slope limiters are still sought. Copyright © 2011 John Wiley & Sons, Ltd.     

An efficient implicit mesh‐less method for compressible flow calculations

A dual‐time implicit mesh‐less scheme is presented for calculation of compressible inviscid flow equations. The Taylor series least‐square method is used for approximation of spatial derivatives at each node which leads to a central difference discretization. Several convergence acceleration techniques such as local time stepping, enthalpy damping and residual smoothing are adopted in this approach. The capabilities of the method are demonstrated by flow computations around single and multi‐element airfoils at subsonic, transonic and supersonic flow conditions. Results are presented which indicate good agreements with other reliable finite‐volume results. The computational time is considerably reduced when using the proposed mesh‐less method compared with the explicit mesh‐less and finite‐volume schemes using the same point distributions. Copyright © 2010 John Wiley & Sons, Ltd.     

Improved third‐order weighted essentially nonoscillatory scheme

A new third‐order WENO scheme is proposed to achieve the desired order of convergence at the critical points for scalar hyperbolic equations. A new reference smoothness indicator is introduced, which satisfies the sufficient condition on the weights for the third‐order convergence. Following the truncation error analysis, we have shown that the proposed scheme achieves the desired order accurate for smooth solutions with arbitrary number of vanishing derivatives if the parameter ε satisfies certain conditions. We have made a comparative study of the proposed scheme with the existing schemes such as WENO‐JS, WENO‐Z, and WENO‐N3 through different numerical examples. The result shows that the proposed scheme (WENO‐MN3) achieves better performance than these schemes.