A high-order accurate hybrid scheme using a central flux scheme and a WENO scheme for compressible flowfield analysis

A high-order accurate hybrid central-WENO scheme is proposed. The fifth order WENO scheme [G.S. Jiang, C.W. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys. 126 (1996) 202–228] is divided into two parts, a central flux part and a numerical dissipation part, and is coupled with a central flux scheme. Two sub-schemes, the WENO scheme and the central flux scheme, are hybridized by means of a weighting function that indicates the local smoothness of the flowfields. The derived hybrid central-WENO scheme is written as a combination of the central flux scheme and the numerical dissipation of the fifth order WENO scheme, which is controlled adaptively by a weighting function. The structure of the proposed hybrid central-WENO scheme is similar to that of the YSD-type filter scheme [H.C. Yee, N.D. Sandham, M.J. Djomehri, Low-dissipative high-order shock-capturing methods using characteristic-based filters, J. Comput. Phys. 150 (1999) 199–238]. Therefore, the proposed hybrid scheme has also certain merits that the YSD-type filter scheme has. The accuracy and efficiency of the developed hybrid central-WENO scheme are investigated through numerical experiments on inviscid and viscous problems. Numerical results show that the proposed hybrid central-WENO scheme can resolve flow features extremely well.     

On high-order shock-fitting and front-tracking schemes for numerical simulation of shock–disturbance interactions

High-order methods that can resolve interactions of flow-disturbances with shock waves are critical for reliable numerical simulation of shock wave and turbulence interaction. Such problems are not well understood due to the limitations of numerical methods. Most of the popular shock-capturing methods are only first-order accurate at the shock and may incur spurious numerical oscillations near the shock. Shock-fitting algorithms have been proposed as an alternative which can achieve uniform high-order accuracy and can avoid possible spurious oscillations incurred in shock-capturing methods by treating shocks as sharp interfaces. We explore two ways for shock-fitting: conventional moving grid set-up and a new fixed grid set-up with front tracking. In the conventional shock-fitting method, a moving grid is fitted to the shock whereas in the newly developed fixed grid set-up the shock front is tracked using Lagrangian points and is free to move across the underlying fixed grid. Different implementations of shock-fitting methods have been published in the literature. However, uniform high-order accuracy of various shock-fitting methods has not been systematically established. In this paper, we carry out a rigorous grid-convergence analysis on different variations of shock-fitting methods with both moving and fixed grids. These shock-fitting methods consist of different combinations of numerical methods for computing flow away from the shock and those for computing the shock movement. Specifically, we consider fifth-order upwind finite-difference scheme and shock-capturing WENO schemes with conventional shock-fitting and show that a fifth-order convergence is indeed achieved for a canonical one-dimensional shock-entropy wave interaction problem. We also show that the method of finding shock velocity from one characteristic relation and Rankine–Hugoniot jump condition performs better than the other methods of computing shock velocities. A high-order front-tracking implementation of shock-fitting is also presented in this paper and nominal rate of convergence is shown. The front-tracking results are validated by comparing to results from the conventional shock-fitting method and a linear-interaction analysis for a two-dimensional shock disturbance interaction problem.     

High-order well-balanced schemes and applications to non-equilibrium flow

The appearance of the source terms in modeling non-equilibrium flow problems containing finite-rate chemistry or combustion poses additional numerical difficulties beyond that for solving non-reacting flows. A well-balanced scheme, which can preserve certain non-trivial steady state solutions exactly, may help minimize some of these difficulties. In this paper, a simple one-dimensional non-equilibrium model with one temperature is considered. We first describe a general strategy to design high-order well-balanced finite-difference schemes and then study the well-balanced properties of the high-order finite-difference weighted essentially non-oscillatory (WENO) scheme, modified balanced WENO schemes and various total variation diminishing (TVD) schemes. The advantages of using a well-balanced scheme in preserving steady states and in resolving small perturbations of such states will be shown. Numerical examples containing both smooth and discontinuous solutions are included to verify the improved accuracy, in addition to the well-balanced behavior.     

Generalized conservative approximations of split convective derivative operators

We propose a strategy to design locally conservative finite-difference approximations of convective derivatives for shock-free compressible flows with arbitrary order of accuracy, that generalizes the approach of Ducros et al. (2000) [1], and that can be applied as a building block of low-dissipative, hybrid shock-capturing methods. The approximations stem from application of standard central difference formulas to split forms of the convective terms in the compressible Euler equations, which guarantee strong numerical stability and (near) energy preservation in the inviscid limit. A convenient implementation of the high-order fluxes is suggested, which guarantees improved computational efficiency over existing methods. Numerical tests performed for isotropic turbulence at zero viscosity show stability of schemes with order of accuracy up to ten, and effectiveness of convective splitting of Kennedy and Gruber (2008) [2] in providing extra stability in the presence of strong density variations. Numerical simulations of compressible turbulent boundary layer flow indicate suitability of the method for non-uniform grids, and overall support superior computational efficiency of high-order schemes.     

A fourth-order divergence-free method for MHD flows

This paper extends our previous third-order method [S. Li, High order central scheme on overlapping cells for magneto-hydrodynamic flows with and without constrained transport method, J. Comput. Phys. 227 (2008) 7368–7393] to the fourth-order. Central finite-volume schemes on overlapping grid are used for both the volume-averaged variables and the face-averaged magnetic field. The magnetic field at the cell boundaries falls within the dual grid and is naturally continuous so that our method eliminates the instability triggered by the discontinuity in the normal component of the magnetic field. Our fourth-order scheme has much smaller numerical dissipation than the third-order scheme. The divergence-free condition of the magnetic field is preserved by our fourth-order divergence-free reconstruction and the constrained transport method. Numerical examples show that the divergence-free condition is essential to the accuracy of the method when a limiter is used in the reconstruction. The high-order, low-dissipation, and divergence-free properties of this method make it an ideal tool for direct magneto-hydrodynamic turbulence simulations.     

A third-order finite-volume residual-based scheme for the 2D Euler equations on unstructured grids

A residual-based (RB) scheme relies on the vanishing of residual at the steady-state to design a transient first-order dissipation, which becomes high-order at steady-state. Initially designed within a finite-difference framework for computations of compressible flows on structured grids, the RB schemes displayed good convergence, accuracy and shock-capturing properties which motivated their extension to unstructured grids using a finite volume (FV) method. A second-order formulation of the FV–RB scheme for compressible flows on general unstructured grids was presented in a previous paper. The present paper describes the derivation of a third-order FV–RB scheme and its application to hyperbolic model problems as well as subsonic, transonic and supersonic internal and external inviscid flows.     

双曲守恒律系统的分段边界层函数方法(PBLM)

本文提出了一个新的高阶Godunov格式。此格式放弃了[9]、[10]中关于参数在格子中满足多项式分布近似及在格子边界上存在间断的假设,直接引入了一个分段边界层型函数分布假设。由于引入的分布函数是单调可微的,因此PBLM格式无需进行如同MUSCL、PPM等格式中的单调性校核。该格式由于不进行单调性修正,在PPM格式中需进行修正而精度降阶的点上仍保持原有精度。对一维激波管的计算表明PBLM格式对激波的展开比PPM格式还要小,计算时间相当。同PPM格式一样,PBLM格式在激波后存在有2%~4%的皮后伪振荡,应加上适当的人工粘性。     

A high-resolution,hybrid compact-WENO scheme with minimized dispersion and controllable dissipation

In this paper,a high-resolution,hybrid compact-WENO scheme is developed based on the minimized dispersion and controllable dissipation reconstruction technique.Firstly,a suffcient condition for a family of tri-diagonal compact schemes to have independent dispersion and dissipation is derived.Then,a specific 4th order compact scheme with low dispersion and adjustable dissipation is constructed and analyzed.Finally,the optimized compact scheme is blended with the WENO scheme to form the hybrid scheme.Moreover,the approximation dispersion relation approach is employed to optimize the spectral properties of the nonlinear scheme to yield the true wave propagation behavior of the finite difference scheme.Several test cases are carried out to verify the highresolution as well as the robust shock-capturing capabilities of the proposed scheme.     

A shock-capturing methodology based on adaptative spatial filtering for high-order non-linear computations

A shock-capturing methodology is developed for non-linear computations using low-dissipation schemes and centered finite differences. It consists in applying an adaptative second-order filtering to handle discontinuities in combination with a background selective filtering to remove grid-to-grid oscillations. The shock-capturing filtering is written in its conservative form, and its magnitude is determined dynamically from the flow solutions. A shock-detection procedure based on a Jameson-like shock sensor is derived so as to apply the shock-capturing filtering only around shocks. A second-order filter with reduced errors in the Fourier space with respect to the standard second-order filter is also designed. Linear and non-linear 1D and 2D problems are solved to show that the methodology is capable of capturing shocks without providing dissipation outside shocks. The shock detection allows in particular to distinguish shocks from linear waves, and from vortices when it is performed from dilatation rather than from pressure. Finally the methodology is simple to implement and reasonable in terms of computational cost.     

Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves

Flows in which shock waves and turbulence are present and interact dynamically occur in a wide range of applications, including inertial confinement fusion, supernovae explosion, and scramjet propulsion. Accurate simulations of such problems are challenging because of the contradictory requirements of numerical methods used to simulate turbulence, which must minimize any numerical dissipation that would otherwise overwhelm the small scales, and shock-capturing schemes, which introduce numerical dissipation to stabilize the solution. The objective of the present work is to evaluate the performance of several numerical methods capable of simultaneously handling turbulence and shock waves. A comprehensive range of high-resolution methods (WENO, hybrid WENO/central difference, artificial diffusivity, adaptive characteristic-based filter, and shock fitting) and suite of test cases (Taylor–Green vortex, Shu–Osher problem, shock-vorticity/entropy wave interaction, Noh problem, compressible isotropic turbulence) relevant to problems with shocks and turbulence are considered. The results indicate that the WENO methods provide sharp shock profiles, but overwhelm the physical dissipation. The hybrid method is minimally dissipative and leads to sharp shocks and well-resolved broadband turbulence, but relies on an appropriate shock sensor. Artificial diffusivity methods in which the artificial bulk viscosity is based on the magnitude of the strain-rate tensor resolve vortical structures well but damp dilatational modes in compressible turbulence; dilatation-based artificial bulk viscosity methods significantly improve this behavior. For well-defined shocks, the shock fitting approach yields good results.     

Compact finite volume schemes on boundary-fitted grids

The paper focuses on the development of a framework for high-order compact finite volume discretization of the three-dimensional scalar advection–diffusion equation. In order to deal with irregular domains, a coordinate transformation is applied between a curvilinear, non-orthogonal grid in the physical space and the computational space. Advective fluxes are computed by the fifth-order upwind scheme introduced by Pirozzoli [S. Pirozzoli, Conservative hybrid compact-WENO schemes for shock–turbulence interaction, J. Comp. Phys. 178 (2002) 81] while the Coupled Derivative scheme [M.H. Kobayashi, On a class of Padé finite volume methods, J. Comp. Phys. 156 (1999) 137] is used for the discretization of the diffusive fluxes.Numerical tests include unsteady diffusion over a distorted grid, linear free-surface gravity waves in a irregular domain and the advection of a scalar field. The proposed methodology attains high-order formal accuracy and shows very favorable resolution characteristics for the simulation of problems with a wide range of length scales.     

A class of finite difference schemes with low dispersion and controllable dissipation for DNS of compressible turbulence

In this paper, a class of finite difference schemes which achieves low dispersion and controllable dissipation in smooth region and robust shock-capturing capabilities in the vicinity of discontinuities is presented. Firstly, a sufficient condition for semi-discrete finite difference schemes to have independent dispersion and dissipation is derived. This condition enables a novel approach to separately optimize the dissipation and dispersion properties of finite difference schemes and a class of schemes with minimized dispersion and controllable dissipation is thus obtained. Secondly, for the purpose of shock-capturing, one of these schemes is used as the linear part of the WENO scheme with symmetrical stencils to constructed an improved WENO scheme. At last, the improved WENO scheme is blended with its linear counterpart to form a new hybrid scheme for practical applications. The proposed scheme is accurate, flexible and robust. The accuracy and resolution of the proposed scheme are tested by the solutions of several benchmark test cases. The performance of this scheme is further demonstrated by its application in the direct numerical simulation of compressible turbulent channel flow between isothermal walls.     

Stability criteria for hybrid difference methods

The stability of hybrid difference methods, where different schemes are used in different parts of the domain, is examined for general schemes. It is shown that the energy method with the natural norm does not prove stability, but that the Kreiss or ‘GKS’ theory yields sufficient criteria for stability. While the analysis is general, it is discussed primarily in the context of hybrid schemes for shock/turbulence interactions, where a robust shock-capturing scheme is used around the discontinuities and an efficient linear scheme is used in other regions. An example of two coupled schemes that are individually stable yet unstable when coupled is given, showing that stability of hybrid methods is an important and non-trivial matter.     

低阶格式在大涡模拟计算中的适用性

采用三维Taylor-Green涡作为研究对象,利用工程中常用的低阶数值格式,研究格式本身的数值误差对大涡模拟计算的影响.结果表明:三种数值格式的数值耗散行为都与亚格子模型行为类似,即在小雷诺数下,流场比较光滑时,耗散很小,当雷诺数增加,流动转捩为湍流,流场梯度增大,耗散显著增大.对于MUSCL格式和二阶有界中心格式,在高雷诺数下,亚格子尺度模型没有明显改善计算结果,但也没有使计算结果恶化.中心格式相比其它两种格式,数值耗散最小,但是在高雷诺数湍流情况下,中心格式的数值耗散仍然主导了能量的耗散,再添加亚格子模型,计算结果反而变得稍差.对于工程中的低阶格式而言,采用中心格式计算大涡模拟是比较好的选择,而且在计算不存在稳定性问题时,采用不添加亚格子模型的隐式大涡模拟效果更好.     

有限差分格式的一个通用色散-耗散条件

通过分析显式有限差分格式的数值色散和数值耗散,导出一个适于有限差分格式的通用色散-耗散条件.根据群速度和耗散率之间的物理关系,确定了用以抑制数值解中伪高波数波所需要的适度耗散.在以往发展的低耗散加权基本无振荡格式WENO-CU6-M2上的应用表明,该条件可用作优化线性或非线性有限差分格式的色散和耗散的通用指导准则.此外,满足色散-耗散条件的改进WENO-CU6-M2格式还可选作低分辨率数值模拟,以三维Taylor-Green涡向湍流转捩和自相似能量衰减问题展现了它的这种能力.与经典的动态Smagorinsky亚网格尺度模型相比,在Reynolds数Re=400~3000条件下,无黏和黏性Taylor-Green涡的数值模拟结果均得到明显改善.在保持激波捕捉特性同时,与最新的隐式大涡模拟模型的计算效果相当.      

An adaptive central-upwind weighted essentially non-oscillatory scheme

In this work, an adaptive central-upwind 6th-order weighted essentially non-oscillatory (WENO) scheme is developed. The scheme adapts between central and upwind schemes smoothly by a new weighting relation based on blending the smoothness indicators of the optimal higher order stencil and the lower order upwind stencils. The scheme achieves 6th-order accuracy in smooth regions of the solution by introducing a new reference smoothness indicator. A number of numerical examples suggest that the present scheme, while preserving the good shock-capturing properties of the classical WENO schemes, achieves very small numerical dissipation.     

The constrained reinitialization equation for level set methods

Based on the constrained reinitialization scheme [D. Hartmann, M. Meinke, W. Schröder, Differential equation based constrained reinitialization for level set methods, J. Comput. Phys. 227 (2008) 6821–6845] a new constrained reinitialization equation incorporating a forcing term is introduced. Two formulations for high-order constrained reinitialization (HCR) are presented combining the simplicity and generality of the original reinitialization equation [M. Sussman, P. Smereka, S. Osher, A level set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys. 114 (1994) 146–159] in terms of high-order standard discretization and the accuracy of the constrained reinitialization scheme in terms of interface displacement. The novel HCR schemes represent simple extensions of standard implementations of the original reinitialization equation. The results evidence the significantly increased accuracy and robustness of the novel schemes.     

采用Power限制器的PNND和PWNND格式

为高精度捕捉激波等流场结构,引入一种Power限制器,对NND格式和WNND格式进行改进,分别得到二阶PNND(Power NND)格式和三阶PWNND(Power WNND)格式.该Power类型格式通过Power限制器对相邻待选模板上的一阶导数进行限制,改善了NND格式和WNND格式在间断附近的耗散效应.对各种格式的分析表明,在间断附近采用Power限制器的格式比原格式的表现要好,耗散小且捕捉间断精度高,其中PNND格式虽然只有二阶精度,但在所有算例中与三阶WNND格式的计算结果比较接近,在个别算例中甚至优于WNND格式.最后将PWNND格式应用到二维NACA0012翼型的强迫俯仰振动的数值模拟,计算结果与实验值、参考计算值吻合较好.     

Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws

In this article, we propose a new class of finite volume schemes of arbitrary accuracy in space and time for systems of hyperbolic balance laws with stiff source terms. The new class of schemes is based on a three stage procedure. First a high-order WENO reconstruction procedure is applied to the cell averages at the current time level. Second, the temporal evolution of the reconstruction polynomials is computed locally inside each cell using the governing equations. In the original ENO scheme of Harten et al. and in the ADER schemes of Titarev and Toro, this time evolution is achieved via a Taylor series expansion where the time derivatives are computed by repeated differentiation of the governing PDE with respect to space and time, i.e. by applying the so-called Cauchy–Kovalewski procedure. However, this approach is not able to handle stiff source terms. Therefore, we present a new strategy that only replaces the Cauchy–Kovalewski procedure compared to the previously mentioned schemes. For the time-evolution part of the algorithm, we introduce a local space–time discontinuous Galerkin (DG) finite element scheme that is able to handle also stiff source terms. This step is the only part of the algorithm which is locally implicit. The third and last step of the proposed ADER finite volume schemes consists of the standard explicit space–time integration over each control volume, using the local space–time DG solutions at the Gaussian integration points for the intercell fluxes and for the space–time integral over the source term. We will show numerical convergence studies for nonlinear systems in one space dimension with both non-stiff and with very stiff source terms up to sixth order of accuracy in space and time. The application of the new method to a large set of different test cases is shown, in particular the stiff scalar model problem of LeVeque and Yee [R.J. LeVeque, H.C. Yee, A study of numerical methods for hyperbolic conservation laws with stiff source terms, Journal of Computational Physics 86 (1) (1990) 187–210], the relaxation system of Jin and Xin [S. Jin, Z. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Communications on Pure and Applied Mathematics 48 (1995) 235–277] and the full compressible Euler equations with stiff friction source terms.