High‐order k‐exact WENO finite volume schemes for solving gas dynamic Euler equations on unstructured grids

This paper presents a family of High‐order finite volume schemes applicable on unstructured grids. The k‐exact reconstruction is performed on every control volume as the primary reconstruction. On a cell of interest, besides the primary reconstruction, additional candidate reconstruction polynomials are provided by means of very simple and efficient ‘secondary’ reconstructions. The weighted average procedure of the WENO scheme is then applied to the primary and secondary reconstructions to ensure the shock‐capturing capability of the scheme. This procedure combines the simplicity of the k‐exact reconstruction with the robustness of the WENO schemes and represents a systematic and unified way to construct High‐order accurate shock capturing schemes. To further improve the efficiency, an efficient problem‐independent shock detector is introduced. Several test cases are presented to demonstrate the accuracy and non‐oscillation property of the proposed schemes. The results show that the proposed schemes can predict the smooth solutions with uniformly High‐order accuracy and can capture the shock waves and contact discontinuities in high resolution. Copyright © 2011 John Wiley & Sons, Ltd.     

求解可压缩流的二维通量分裂格式

传统的一维通量分裂格式在计算界面数值通量时,只考虑网格界面法向的波系。采用传统的TV格式分别求解对流通量和压力通量。通过求解考虑了横向波系影响的角点数值通量来构造一种真正二维的TV通量分裂格式。在计算一维数值算例时,该格式与传统的TV格式具有相同的数值通量计算公式,因此其保留了传统的TV格式精确捕捉接触间断和膨胀激波的优点。在计算二维算例时,该格式比传统的TV格式具有更高的分辨率;在计算二维强激波问题时,消除了传统TV格式的非物理现象,表现出更好的鲁棒性;此外,该格式大大提高了稳定性CFL数,从而具有更高的计算效率。因此,本文方法是一种精确、高效并且具有强鲁棒性的数值方法,在可压缩流的数值模拟中具有广阔的应用前景。     

Kinetic theory based multi-level adaptive finite difference WENO schemes for compressible Euler equations

In this paper we proposed the kinetic framework based fifth-order adaptive finite difference WENO schemes abbreviated as WENO-AO-K schemes to solve the compressible Euler equations, which are quasi-linear hyperbolic equations that can admit discontinuous solutions like shock and contact waves. The formulation of the proposed schemes is based on the kinetic theory where one can recover the Euler equations by applying a suitable moment method strategy to the Boltzmann equation. The kinetic flux vector splitting strategy is used in WENO-AO framework, which produces the computationally expensive error and exponential functions. Thus, to reduce the computational cost, a physically more relevant peculiar velocity based splitting strategy is used, which is more efficient than the kinetic flux vector splitting. High order of accuracy in time is achieved using the third-order total variation diminishing Runge–Kutta (TVD-RK) scheme. Several one- and two-dimensional test cases are solved for the compressible Euler equations using the proposed fifth-order WENO-AO-K schemes and the results are compared with conventional WENO-AO scheme. Proposed schemes capture the complex flow features in a smooth region accurately, and discontinuity is also well resolved. Error analysis of the proposed schemes shows optimal convergence rates in various norms.     

An arbitrary Lagrangian–Eulerian method based on the adaptive Riemann solvers for general equations of state

Approximate or exact Riemann solvers play a key role in Godunov‐type methods. In this paper, three approximate Riemann solvers, the MFCAV, DKWZ and weak wave approximation method schemes, are investigated through numerical experiments, and their numerical features, such as the resolution for shock and contact waves, are analyzed and compared. Based on the analysis, two new adaptive Riemann solvers for general equations of state are proposed, which can resolve both shock and contact waves well. As a result, an ALE method based on the adaptive Riemann solvers is formulated. A number of numerical experiments show good performance of the adaptive solvers in resolving both shock waves and contact discontinuities. Copyright © 2008 John Wiley & Sons, Ltd.     

有限谱ENO格式及其应用

首先对王健平提出的有限谱法^[1-3]做了进一步的理论研究，发现了一些新的有限谱法的插值基函数组，并互将有限谱法应用在ENO格式中，构造了有限谱ENO计算格式，然后通过对一维Euler方程的几个经典的模型问题和二维湍流与弱激波相互作用问题的数值计算，并且与理论解或准精确解进行比较分析，从而表明了此格式对于激波和其他间断具有较高的分辨率，在激波附近基本上没有明显的数值振荡，而且对于流场中的细致结构也具有相当高的精度。     

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.     

Efficient high‐resolution relaxation schemes for hyperbolic systems of conservation laws

In this work, we present a total variation diminishing (TVD) scheme in the zero relaxation limit for nonlinear hyperbolic conservation law using flux limiters within the framework of a relaxation system that converts a nonlinear conservation law into a system of linear convection equations with nonlinear source terms. We construct a numerical flux for space discretization of the obtained relaxation system and modify the definition of the smoothness parameter depending on the direction of the flow so that the scheme obeys the physical property of hyperbolicity. The advantages of the proposed scheme are that it can give second‐order accuracy everywhere without introducing oscillations for 1‐D problems (at least with) smooth initial condition. Also, the proposed scheme is more efficient as it works for any non‐zero constant value of the flux limiter ? ? [0, 1], where other TVD schemes fail. The resulting scheme is shown to be TVD in the zero relaxation limit for 1‐D scalar equations. Bound for the limiter function is obtained. Numerical results support the theoretical results. Copyright © 2007 John Wiley & Sons, Ltd.     

RCM‐TVD hybrid scheme for hyperbolic conservation laws

We describe a hybrid method for the solution of hyperbolic conservation laws. A third‐order total variation diminishing (TVD) finite difference scheme is conjugated with a random choice method (RCM) in a grid‐based adaptive way. An efficient multi‐resolution technique is used to detect the high gradient regions of the numerical solution in order to capture the shock with RCM while the smooth regions are computed with the more efficient TVD scheme. The hybrid scheme captures correctly the discontinuities of the solution and saves CPU time. Numerical experiments with one‐ and two‐dimensional problems are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

群速度控制格式及二维Riemann解

强激波和强接触间断的数值模拟一直是计算流体力学里一个富有挑战性的课题,它们是很多实际流动的基础。三阶迎风紧致格式是一种具有较高分辨率的高精度方法,但是在计算激波时仍有数值振荡产生。本文根据数值解的群速度特性,在三阶迎风紧致格式的基础上提出了一种群速度控制格式,使得能够正确模拟含有强激波和强接触间断的复杂流动。在此基础上构造了求解包含大压力比和密度比的二维界面问题的数值方法。计算结果表明,方法对激波和接触间断的分辨效果是令人满意的。     

Improvement of multistep WENO scheme and its extension to higher orders of accuracy

This paper presents an efficient procedure for overcoming the deficiency of weighted essentially non‐oscillatory schemes near discontinuities. Through a thorough incorporation of smoothness indicators into the weights definition, up to ninth‐order accurate multistep methods are devised, providing weighted essentially non‐oscillatory schemes with enhanced order of convergence at transition points from smooth regions to a discontinuity, while maintaining stability and the essentially non‐oscillatory behavior. We also provide a detailed analysis of the resolution power and show that the solution enhancements of the new method at smooth regions come from their ability to render smoothness indicators closer to uniformity. The new scheme exhibits similar fidelity as other multistep schemes; however, with superior characteristics in terms of robustness and efficiency, as no logical statements or mapping function is needed. Extensions to higher orders of accuracy present no extra complexity. Numerical solutions of linear advection problems and nonlinear hyperbolic conservation laws are used to demonstrate the scheme's improved behavior for shock‐capturing problems. Copyright © 2016 John Wiley & Sons, Ltd.     

A hybrid kinetic WENO scheme for inviscid and viscous flows

In this paper, we propose a high‐order finite volume hybrid kinetic Weighted Essentially Non‐Oscillatory (WENO) scheme for inviscid and viscous flows. Based on the WENO reconstruction technique, a hybrid kinetic numerical flux is introduced for the present method, which includes the mechanisms of both the free transfer and the collision of gas molecules. The collisionless free transfer part of the hybrid numerical flux is constructed from the conventional kinetic flux vector splitting treatment, and the collision contribution is considered by constructing an equilibrium gas state and calculating the corresponding numerical flux at the cell interface. The total variation diminishing Runge–Kutta methods are used for the temporal integration. The high‐order accuracy and good shock‐capturing capability of the proposed hybrid kinetic WENO scheme are validated by many numerical examples in one‐dimensional and two‐dimensional cases. Copyright © 2015 John Wiley & Sons, Ltd.     

Realization of contact resolving approximate Riemann solvers for strong shock and expansion flows

The Harten–Lax–van Leer contact (HLLC) and Roe schemes are good approximate Riemann solvers that have the ability to resolve shock, contact, and rarefaction waves. However, they can produce spurious solutions, called shock instabilities, in the vicinity of strong shock. In strong expansion flows, the Roe scheme can admit nonphysical solutions such as expansion shock, and it sometimes fails. We carefully examined both schemes and propose simple methods to prevent such problems. High‐order accuracy is achieved using the weighted average flux (WAF) and MUSCL‐Hancock schemes. Using the WAF scheme, the HLLC and Roe schemes can be expressed in similar form. The HLLC and Roe schemes are tested against Quirk's test problems, and shock instability appears in both schemes. To remedy shock instability, we propose a control method of flux difference across the contact and shear waves. To catch shock waves, an appropriate pressure sensing function is defined. Using the proposed method, shock instabilities are successfully controlled. For the Roe scheme, a modified Harten–Hyman entropy fix method using Harten–Lax–van Leer‐type switching is suggested. A suitable criterion for switching is established, and the modified Roe scheme works successfully with the suggested method. Copyright © 2009 John Wiley & Sons, Ltd.     

一种健壮的混合Roe黎曼求解器

使用Roe格式计算多维流动问题时,在强激波附近会出现数值激波不稳定现象。带有剪切粘性的HLLEC格式不仅可以捕捉接触间断,而且表现出很好的稳定性。混合Roe格式和HLLEC格式来消除数值激波不稳定性。在强激波附近,通过激波面法向和网格界面法向的夹角来定义开关函数,使得数值通量在激波面横向切换成HLLEC格式。在其余地方,数值通量依然使用Roe格式来计算。数值试验表明,混合格式不仅消除了Roe格式的数值激波不稳定性,还最大程度地减少了HLLEC格式所带来的剪切耗散,保留了Roe格式高分辨率的优点。     

Discontinuity-capturing finite element computation of unsteady flow with adaptive unstructured mesh

A discontinuity-capturing scheme of finite element method (FEM) is proposed. The unstructured-grid technique combined with a new type of adaptive mesh approach is developed for both compressible and incompressible unsteady flows, which exhibits the capability of capturing the shock waves and/or thin shear layers accurately in an unsteady viscous flow at high Reynolds number. In particular, a new testing variable, i.e., the disturbed kinetic energyE, is suggested and used in the adaptive mesh computation, which is universally applicable to the capturing of both shock waves and shear layers in the inviscid flow and viscous flow at high Reynolds number. Based on several calculated examples, this approach has been proved to be effective and efficient for the calculations of compressible and incompressible flows. The project supported by the National Natural Science Foundation of China (10125210), the Hundred-Talent Programme of the Chinese Academy of Sciences and the Innovation Project of the Chinese Academy of Sciences (KJCX-SW-L04, KJCX2-SW-L2)     

一种健壮的TV通量分裂格式

随着计算流体力学的快速发展,设计精确、高效并且健壮的数值格式变得尤为重要。Toro等^[8]提出的TV通量分裂格式表现出简单、高效和精确分辨接触间断等优点,但是在计算一些多维算例时会出现数值激波不稳定现象。两波近似的HLL格式在计算中非常高效和健壮,但是不能分辨接触间断大大地限制了其应用。本文对TV通量分裂格式进行稳定性分析,据此提出一种混合格式来消除TV格式的数值激波不稳定性。数值试验表明,本文构造的混合格式不仅保留了原始TV格式的优点,而且具有更好的健壮性,在计算二维问题时不会出现数值激波不稳定现象。     

High resolution Godunov‐type schemes with small stencils

Higher‐order Godunov‐type schemes have to cope with the following two problems: (i) the increase in the size of the stencil that make the scheme computationally expensive, and (ii) the monotony‐preserving treatments (limiters) that must be implemented to avoid oscillations, leading to strong damping of the solution, in particular linear waves (e.g. acoustic waves). When too compressive, limiting procedures may also trigger the instability of oscillatory numerical solutions (e.g. in advection–dispersion phenomena) via the artificial amplification of the shorter modes. The present paper proposes a new approach to carry out the reconstruction. In this approach, the values of the flow variable at the edges of the computational cells are obtained directly from the reconstruction within these cells. This method is applied to the MUSCL and DPM schemes for the solution of the linear advection equation. The modified DPM scheme can capture contact discontinuities within one computational cell, even after millions of time steps at Courant numbers ranging from 1 to values as low as 10^‐4. Linear waves are subject to negligible damping. Application of the method to the DPM for one‐dimensional advection–dispersion problems shows that the numerical instability of oscillatory solutions caused by the over compressive, original DPM limiter is eliminated. One‐ and two‐dimensional shallow water simulations show an improvement over classical methods, in particular for two‐dimensional problems with strongly distorted meshes. The quality of the computational solution in the two‐dimensional case remains acceptable even for mesh aspect ratios Δx/Δy as large as 10. The method can be extend to the discretization of higher‐order PDEs, allowing third‐order space derivatives to be discretized using only two cells in space. Copyright © 2004 John Wiley & Sons, Ltd.     

Developing a hybrid flux function suitable for hypersonic flow simulation with high‐order methods

In this paper, we develop a new hybrid Euler flux function based on Roe's flux difference scheme, which is free from shock instability and still preserves the accuracy and efficiency of Roe's flux scheme. For computational cost, only 5% extra CPU time is required compared with Roe's FDS. In hypersonic flow simulation with high‐order methods, the hybrid flux function would automatically switch to the Rusanov flux function near shock waves to improve the robustness, and in smooth regions, Roe's FDS would be recovered so that the advantages of high‐order methods can be maintained. Multidimensional dissipation is introduced to eliminate the adverse effects caused by flux function switching and further enhance the robustness of shock‐capturing, especially when the shock waves are not aligned with grids. A series of tests shows that this new hybrid flux function with a high‐order weighted compact nonlinear scheme is not only robust for shock‐capturing but also accurate for hypersonic heat transfer prediction. Copyright © 2015 John Wiley & Sons, Ltd.     

Evaluation of adaptive mesh refinement and coarsening for the computation of compressible flows on unstructured meshes

The compressible gas flows of interest to aerospace applications often involve situations where shock and expansion waves are present. Decreasing the characteristic dimension of the computational cells in the vicinity of shock waves improves the quality of the computed flows. This reduction in size may be accomplished by the use of mesh adaption procedures. In this paper an analysis is presented of an adaptive mesh scheme developed for an unstructured mesh finite volume upwind computer code. This scheme is tailored to refine or coarsen the computational mesh where gradients of the flow properties are respectively high or low. The refinement and coarsening procedures are applied to the classical gas dynamic problems of the stabilization of shock waves by solid bodies. In particular, situations where oblique shock waves interact with an expansion fan and where bow shocks arise around solid bodies are considered. The effectiveness of the scheme in reducing the computational time, while increasing the solution accuracy, is assessed. It is shown that the refinement procedure alone leads to a number of computational cells which is 20% larger than when alternate passes of refinement and coarsening are used. Accordingly, a reduction of computational time of the same order of magnitude is obtained. Copyright © 2005 John Wiley & Sons, Ltd.     

A 3D UNSTRUCTURED MESH ADAPTATION ALGORITHM FOR TIME-DEPENDENT SHOCK-DOMINATED PROBLEMS

In this paper we present a tetrahedron-based, h-refinement-type algorithm for the solution of problems in 3D gas dynamics using unstructured mesh adaptation. The mesh adaptation algorithm is coupled to a cell-centred, Riemann problem-based, finite volume scheme of the MUSCL type, employing an approximate Riemann solver. The adaptive scheme is then used to compute the diffraction of shock waves around a box section corner for subsonic and supersonic post-shock flow. In the subsonic case, preliminary measurements of vortex filament speed and vortical Mach number are in broad quantitative agreement with known theoretical results. © 1997 John Wiley & Sons, Ltd.     

Review of Computational Aeroacoustics Algorithms

This paper presents a review of recent advancements in computational methodology for aeroacoustics problems. High-order finite difference methods for computation of linear and nonlinear acoustic waves are the primary focus of the review. Schemes for numerical simulation of linear waves include explicit optimized and DRP finite-difference operators, compact schemes, wavenumber extended upwind schemes and leapfrog-like algorithms. Both spatial approximations and time-integration techniques, which include low-dissipation low-dispersion Adams-Bashforth and Runge-Kutta (RK) methods, are examined. Wave propagation properties are analysed in the wavenumber and frequency space. Different approaches to eliminate short-wave spurious numerical waves are also reviewed. Methods for simulating nonlinear acoustic phenomena include essentially non-oscillatory (ENO) schemes, numerical adaptive filtering for high-order explicit and compact finite-difference operators, MacCormack and adaptive compact nonlinear algorithms. A literature survey of other CAA methods is provided in the introductory part.