High-order Finite Difference and Finite Volume WENO Schemes and Discontinuous Galerkin Methods for CFD

In recent years, high order numerical methods have been widely used in computational fluid dynamics (CFD), to effectively resolve complex flow features using meshes which are reasonable for today's computers. In this paper, we review and compare three types of high order methods being used in CFD, namely the weighted essentially non-oscillatory (WENO) finite difference methods, the WENO finite volume methods, and the discontinuous Galerkin (DG) finite element methods. We summarize the main features of these methods, from a practical user's point of view, indicate their applicability and relative strength, and show a few selected numerical examples to demonstrate their performance on illustrative model CFD problems.     

High‐order semi‐implicit time‐integrators for a triangular discontinuous Galerkin oceanic shallow water model

We extend the explicit in time high‐order triangular discontinuous Galerkin (DG) method to semi‐implicit (SI) and then apply the algorithm to the two‐dimensional oceanic shallow water equations; we implement high‐order SI time‐integrators using the backward difference formulas from orders one to six. The reason for changing the time‐integration method from explicit to SI is that explicit methods require a very small time step in order to maintain stability, especially for high‐order DG methods. Changing the time‐integration method to SI allows one to circumvent the stability criterion due to the gravity waves, which for most shallow water applications are the fastest waves in the system (the exception being supercritical flow where the Froude number is greater than one). The challenge of constructing a SI method for a DG model is that the DG machinery requires not only the standard finite element‐type area integrals, but also the finite volume‐type boundary integrals as well. These boundary integrals pose the biggest challenge in a SI discretization because they require the construction of a Riemann solver that is the true linear representation of the nonlinear Riemann problem; if this condition is not satisfied then the resulting numerical method will not be consistent with the continuous equations. In this paper we couple the SI time‐integrators with the DG method while maintaining most of the usual attributes associated with DG methods such as: high‐order accuracy (in both space and time), parallel efficiency, excellent stability, and conservation. The only property lost is that of a compact communication stencil typical of time‐explicit DG methods; implicit methods will always require a much larger communication stencil. We apply the new high‐order SI DG method to the shallow water equations and show results for many standard test cases of oceanic interest such as: standing, Kelvin and Rossby soliton waves, and the Stommel problem. The results show that the new high‐order SI DG model, that has already been shown to yield exponentially convergent solutions in space for smooth problems, results in a more efficient model than its explicit counterpart. Furthermore, for those problems where the spatial resolution is sufficiently high compared with the length scales of the flow, the capacity to use high‐order (HO) time‐integrators is a necessary complement to the employment of HO space discretizations, since the total numerical error would be otherwise dominated by the time discretization error. In fact, in the limit of increasing spatial resolution, it makes little sense to use HO spatial discretizations coupled with low‐order time discretizations. Published in 2009 by John Wiley & Sons, Ltd.     

线性与非线性流固耦合动力学数值方法的进展及应用

本文综述了线性与非线性流固耦合问题数值方法的进展及工程应用。讨论了四种数值分析方法：(1)混合有限元–子结构–子区域数值模型,以求解有限域线性流固耦合问题,如流体晃动,声腔–结构耦合,流体中的压力波,化工容器的地震响应,坝水耦合等;(2)混合有限元–边界元数值模型,以求解涉及无限域的线性流固耦合问题,如大型浮体承受飞机降落冲击,船舰的炮击回应等;(3)混合有限元–有限差分(体积)数值模型,以求解不涉及破浪和两相分离的非线性流固耦合问题;(4)混合有限元–光滑粒子数值模型,以求解涉及破浪和两相分离的非线性流固耦合问题。文中推荐分区迭代求解过程,以便应用现有的固体及流体求解器,于毎一时间步长分别求解固体及流体的方程,通过耦合迭代收敛,向前推进以达问题求解。文中选用的工程应用例子包含气–液–壳三相耦合,液化天然气船水晃动,人体步行冲击引起的声腔–建筑结构耦合,大型浮体承受飞机降落冲击的瞬态动力回应,涉及破浪和两相分离的气–翼耦合及结构于水上降落的冲击。数值分析结果与可用的实验或计算结果作了比较,以说明所述方法的精度及工程应用价值。文中列出了基于流固耦合的波能采积装置模型,以应用线性系统的共振及非线性系统的周期解原理,有效地采积波能。本文列出了231篇参考文献,以便读者进一步研讨所感兴趣方法。     

Developments of numerical methods for linear and nonlinear fluid-solid interaction dynamics with applications

本文综述了线性与非线性流固耦合问题数值方法的进展及工程应用. 讨论了四种数值分析方法: (1) 混合有限元-子结构-子区域数值模型, 以求解有限域线性流固耦合问题, 如流体晃动, 声腔-结构耦合, 流体中的压力波, 化工容器的地震响应,坝水耦合等; (2) 混合有限元-边界元数值模型, 以求解涉及无限域的线性流固耦合问题, 如大型浮体承受飞机降落冲击, 船舰的炮击回应等; (3) 混合有限元-有限差分(体积) 数值模型, 以求解不涉及破浪和两相分离的非线性流固耦合问题; (4) 混合有限元-光滑粒子数值模型, 以求解涉及破浪和两相分离的非线性流固耦合问题. 文中推荐分区迭代求解过程, 以便应用现有的固体及流体求解器, 于毎一时间步长分别求解固体及流体的方程, 通过耦合迭代收敛, 向前推进以达问题求解. 文中选用的工程应用例子包含气-液-壳三相耦合, 液化天然气船水晃动, 人体步行冲击引起的声腔-建筑结构耦合, 大型浮体承受飞机降落冲击的瞬态动力回应, 涉及破浪和两相分离的气-翼耦合及结构于水上降落的冲击. 数值分析结果与可用的实验或计算结果作了比较, 以说明所述方法的精度及工程应用价值. 文中列出了基于流固耦合的波能采积装置模型, 以应用线性系统的共振及非线性系统的周期解原理, 有效地采积波能. 本文列出了231 篇参考文献, 以便读者进一步研讨所感兴趣方法.     

Accuracy and efficiency of two numerical methods of solving the potential flow problem for highly nonlinear and dispersive water waves

The accuracy and efficiency of two methods of resolving the exact potential flow problem for nonlinear waves are compared using three different one horizontal dimension (1DH) test cases. The two model approaches use high‐order finite difference schemes in the horizontal dimension and differ in the resolution of the vertical dimension. The first model uses high‐order finite difference schemes also in the vertical, while the second model applies a spectral approach. The convergence, accuracy, and efficiency of the two models are demonstrated as a function of the temporal, horizontal, and vertical resolutions for the following: (1) the propagation of regular nonlinear waves in a periodic domain; (2) the motion of nonlinear standing waves in a domain with fully reflective boundaries; and (3) the propagation and shoaling of a train of waves on a slope. The spectral model approach converges more rapidly as a function of the vertical resolution. In addition, with equivalent vertical resolution, the spectral model approach shows enhanced accuracy and efficiency in the parameter range used for practical model applications. Copyright © 2015 John Wiley & Sons, Ltd.     

Finite element method for viscoelastic flows based on the discrete adaptive viscoelastic stress splitting and the discontinuous Galerkin method: DAVSS-G/DG

Accurate and robust finite element methods for computing flows with differential constitutive equations require approximation methods that numerically preserve the ellipticity of the saddle point problem formed by the momentum and continuity equations and give numerically stable and accurate solutions to the hyperbolic constitutive equation. We present a new finite element formulation based on the synthesis of three ideas: the discrete adaptive splitting method for preserving the ellipticity of the momentum/continuity pair (the DAVSS formulation), independent interpolation of the components of the velocity gradient tensor (DAVSS-G), and application of the discontinuous Galerkin (DG) method for solving the constitutive equation. We call the method DAVSS-G/DG. The DAVSS-G/DG method is compared with several other methods for flow past a cylinder in a channel with the Oldroyd-B and Giesekus constitutive models. Results using the Streamline Upwind Petrov–Galerkin method (SUPG) show that introducing the adaptive splitting increases considerably the range of Deborah number (De) for convergence of the calculations over the well established EVSS-G formulation. When both formulations converge, the DAVSS-G and DEVSS-G methods give comparable results. Introducing the DG method for solution of the constitutive equation extends further the region of convergence without sacrificing accuracy. Calculations with the Oldroyd-B model are only limited by approximation of the almost singular gradients of the axial normal stress that develop near the rear stagnation point on the cylinder. These gradients are reduced in calculations with the Giesekus model. Calculations using the Giesekus model with the DAVSS-G/DG method can be continued to extremely large De and converge with mesh refinement.     

间断Galerkin有限元和有限体积混合计算方法研究

通过局部坐标变换而建立的非正交单元间断Galerkin(DG)有限元计算方法计算精度高， 但计算量大、内存需求大；而非结构网格有限体积方法虽然准确计算热流的问题目 前还没有完全解决，但其具有计算速度快和内存需求小的优点. 该研究是将有 限元和有限体积方法的优点结合，发展有限元和有限体积的混合方法. 在物面 附近黏性占主导作用的区域内采用有限元方法进行计算，在远离物面的区域采用快速的有限 体积方法进行计算，在有限元和有限体积方法结合处要保证通量守恒. 通过算例说明有 限元和有限体积混合方法既能保证黏性区域的热流计算精度和流场结构的分辨率，又能 降低内存需求和提高计算效率，使有限元方法应用于复杂外形(实际工程问题)的计 算成为可能.     

Finite volume methods for unidirectional dispersive wave models

We extend the framework of the finite volume method to dispersive unidirectional water wave propagation in one space dimension. In particular, we consider a KdV–BBM‐type equation. Explicit and implicit–explicit Runge–Kutta‐type methods are used for time discretizations. The fully discrete schemes are validated by direct comparisons to analytic solutions. Invariants' conservation properties are also studied. Main applications include important nonlinear phenomena such as dispersive shock wave formation, solitary waves, and their various interactions. Copyright © 2012 John Wiley & Sons, Ltd.     

基于静动态混合重构的DG/FV混合格式

通过比较紧致格式和间断Galerkin(DG)格式, 提出了``静态重构'和``动态重构'的概念,对有限体积方法和DG有限元方法进行统一的表述. 借鉴有限体积的思想, 发展了基于``混合重构'技术的一类新的DG格式, 称之为间断Galerkin有限元/有限体积混合格式(DG/FV格式). 该类混合格式通过适当地扩展模板(拓展至紧邻单元)重构单元内的高阶多项式分布, 在提高精度的同时, 减少了传统DG格式的计算量和存储量. 通过典型一维和二维标量方程的计算发现新的混合格式在有些情况下具有超收敛(superconvergence)性质.      

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.     

A reconstructed discontinuous Galerkin method for multi-material hydrodynamics with sharp interfaces

Discontinuous Galerkin (DG) methods have been well established for single-material hydrodynamics. However, consistent DG discretizations for non-equilibrium multi-material (more than two materials) hydrodynamics have not been extensively studied. In this work, a novel reconstructed DG (rDG) method for the single-velocity multi-material system is presented. The multi-material system being considered assumes stiff velocity relaxation, but does not assume pressure and temperature equilibrium between the multiple materials. A second-order DG(P₁) method and a third-order least-squares based rDG(P₁P₂) are used to discretize this system in space, and a third-order total variation diminishing (TVD) Runge-Kutta method is used to integrate in time. A well-balanced DG discretization of the non-conservative system is presented and is verified by numerical test problems. Furthermore, a consistent interface treatment is implemented, which ensures strict conservation of material masses and total energy. Numerical tests indicate that the DG and rDG methods are, indeed, the second- and third-order accurate. Comparisons with the second-order finite volume method show that the DG and rDG methods are able to capture the interfaces more sharply. The DG and rDG methods are also more accurate in the single-material regions of the flow. This work focuses on the general multidimensional rDG formulation of the non-equilibrium multi-material system and a study of properties of the method via one-dimensional numerical experiments. The results from this research will be the foundation for a multidimensional high-order rDG method for multi-material hydrodynamics.     

Divergence‐free discontinuous Galerkin schemes for the Stokes equations and the MAC scheme

We show that recently studied discontinuous Galerkin discretizations in their lowest order version are very similar to the marker and cell (MAC) finite difference scheme. Indeed, applying a slight modification, the exact MAC scheme can be recovered. Therefore, the analysis applied to the DG methods applies to the MAC scheme as well and the DG methods provide a natural generalization of the MAC scheme to higher order and irregular meshes. Copyright © 2007 John Wiley & Sons, Ltd.     

Existence and uniqueness of a solution to a problem of steady composite waves on the surface of a liquid flowing over a wavy bottom

Composite waves on the surface of the stationary flow of a heavy ideal incompressible liquid are steady forced waves of finite amplitude which do not disappear when the pressure on the free surface becomes constant but rather are transformed into free nonlinear waves [1]. It will be shown that such waves correspond to the case of nonlinear resonance, and mathematically to the bifurcation of the solution of the fundamental integral equation describing these waves. In [2], a study is made of the problem of composite waves in a flow of finite depth generated by a variable pressure with periodic distribution along the surface of the flow. In [3], such waves are considered for a flow with a wavy bottom. In this case, composite waves are defined as steady forced waves of finite amplitude that, when the pressure becomes constant and the bottom is straightened, do not disappear but are transformed into free nonlinear waves over a flat horizontal bottom. However, an existence and uniqueness theorem was not proved for this case. The aim of the present paper is to fill this gap and investigate the conditions under which such waves can arise.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 88–98, July–August, 1980.     

Provably stable and time‐accurate extensions of Runge–Kutta schemes for CFD computations on moving grids

Extending fixed‐grid time integration schemes for unsteady CFD applications to moving grids, while formally preserving their numerical stability and time accuracy properties, is a nontrivial task. A general computational framework for constructing stability‐preserving ALE extensions of Eulerian multistep time integration schemes can be found in the literature. A complementary framework for designing accuracy‐preserving ALE extensions of such schemes is also available. However, the application of neither of these two computational frameworks to a multistage method such as a Runge–Kutta (RK) scheme is straightforward. Yet, the RK methods are an important family of explicit and implicit schemes for the approximation of solutions of ordinary differential equations in general and a popular one in CFD applications. This paper presents a methodology for filling this gap. It also applies it to the design of ALE extensions of fixed‐grid explicit and implicit second‐order time‐accurate RK (RK2) methods. To this end, it presents the discrete geometric conservation law associated with ALE RK2 schemes and a method for enforcing it. It also proves, in the context of the nonlinear scalar conservation law, that satisfying this discrete geometric conservation law is a necessary and sufficient condition for a proposed ALE extension of an RK2 scheme to preserve on moving grids the nonlinear stability properties of its fixed‐grid counterpart. All theoretical findings reported in this paper are illustrated with the ALE solution of inviscid and viscous unsteady, nonlinear flow problems associated with vibrations of the AGARD Wing 445.6. Copyright © 2011 John Wiley & Sons, Ltd.     

关于气动声学数值计算的方法与进展

气动声学数值计算是近年才出现的研究领域。本文介绍了气动声学数值计算的方法和有关的问题、边界条件的处理以及计算非线性声波的数值方法和进展。讨论了计算气动声学(CAA)的特性及其与计算流体力学(CFD)的差异，指出气动声学数值方法的关键是建立能保持色散关系的差分方程和正确处理无反射边界条件。对于非线性声波传播的问题，为了得到正确的解，应注意提高差分格式对短波的分辨能力，同时发展能抑制“伪”振荡(短波)而对长波基本不起作用的数值方法。     

PERTURBATION FINITE ELEMENT METHOD FOR SOLVING GEOMETRICALLY NONLINEAR PROBLEMS OF AXISYMMETRICAL SHELL

In analysing the geometrically nonlinear problem of an axisymmetrical thin-walled shell, the paper combines the perturbation method with the finite element method by introducing the former into the variational equation to obtain a series of linear equations of different orders and then solving the equations with the latter. It is well-known that the finite element method can be used to deal with difficult problems as in the case of structures with complicated shapes or boundary conditions, and the perturbation method can change the nonlinear problems into linear ones. Evidently the combination of the two methods will give an efficient solution to many difficult nonlinear problems and clear away some obstacles resulted from using any of the two methods solely. The paper derives all the formulas concerning an axisym-metric shell of large deformation by means of the perturbation finite element method and gives two numerical examples,the results of which show good convergence characteristics.     

High-order discontinuous Galerkin method for applications to multicomponent and chemically reacting flows

This article focuses on the development of a discontinuous Galerkin (DG) method for simulations of multicomponent and chemically reacting flows. Compared to aerodynamic flow applications, in which DG methods have been successfully employed, DG simulations of chem-ically reacting flows introduce challenges that arise from flow unsteadiness, combustion, heat release, compressibility effects, shocks, and variations in thermodynamic proper-ties. To address these challenges, algorithms are developed, including an entropy-bounded DG method, an entropy-residual shock indicator, and a new formulation of artificial viscosity. The performance and capabilities of the resulting DG method are demonstrated in several relevant applications, including shock/bubble interaction, turbulent combustion, and detonation. It is concluded that the developed DG method shows promising performance in application to multicompo-nent reacting flows. The paper concludes with a discussion of further research needs to enable the application of DG methods to more complex reacting flows.     

TL法和UL法对刚架弹性大位移分析的适用性

本文分别基于Total Lagrange(TL)和Updated Lagrange(UL)描述分别导出了适用于刚架问题几何非线性分析的有限元公式系统，并研制了相应的计算程序，文中从量化的角度讨论了上述两种列式方法在通常情况下的适用性，并研究了诸如有限单元划分、杆件中轴向力的大小以及采用通常积分方法或积分平均化方法时等多种因素对于有限单元TL法和UL法适用性方面的影响，本文得出了一些相对量化的结论，对于非线性有限元方法的实际工程应用具有一定的指导意义。     

A coupled continuous and discontinuous finite element method for the incompressible flows

In this paper, we develop a coupled continuous Galerkin and discontinuous Galerkin finite element method based on a split scheme to solve the incompressible Navier–Stokes equations. In order to use the equal order interpolation functions for velocity and pressure, we decouple the original Navier–Stokes equations and obtain three distinct equations through the split method, which are nonlinear hyperbolic, elliptic, and Helmholtz equations, respectively. The hybrid method combines the merits of discontinuous Galerkin (DG) and finite element method (FEM). Therefore, DG is concerned to accomplish the spatial discretization of the nonlinear hyperbolic equation to avoid using the stabilization approaches that appeared in FEM. Moreover, FEM is utilized to deal with the Poisson and Helmholtz equations to reduce the computational cost compared with DG. As for the temporal discretization, a second‐order stiffly stable approach is employed. Several typical benchmarks, namely, the Poiseuille flow, the backward‐facing step flow, and the flow around the cylinder with a wide range of Reynolds numbers, are considered to demonstrate and validate the feasibility, accuracy, and efficiency of this coupled method. Copyright © 2016 John Wiley & Sons, Ltd.     

基于非结构/混合网格的高阶精度格式研究进展

尽管以二阶精度格式为基础的计算流体力学(CFD) 方法和软件已经在航空航天飞行器设计中发挥了重要的作用, 但是由于二阶精度格式的耗散和色散较大, 对于湍流、分离等多尺度流动现象的模拟, 现有成熟的CFD 软件仍难以给出满意的结果, 为此CFD 工作者发展了众多的高阶精度计算格式. 如果以适应的计算网格来分类, 一般可以分为基于结构网格的有限差分格式、基于非结构/混合网格的有限体积法和有限元方法,以及各种类型的混合方法. 由于非结构/混合网格具有良好的几何适应性, 基于非结构/混合网格的高阶精度格式近年来备受关注. 本文综述了近年来基于非结构/混合网格的高阶精度格式研究进展, 重点介绍了空间离散方法, 主要包括k-Exact 和ENO/WENO 等有限体积方法, 间断伽辽金(DG) 有限元方法, 有限谱体积(SV) 和有限谱差分(SD) 方法, 以及近来发展的各种DG/FV 混合算法和将各种方法统一在一个框架内的CPR (correctionprocedure via reconstruction) 方法等. 随后简要介绍了高阶精度格式应用于复杂外形流动数值模拟的一些需要关注的问题, 包括曲边界的处理方法、间断侦测和限制器、各种加速收敛技术等. 在综述过程中, 介绍了各种方法的优势与不足, 其间介绍了作者发展的基于"静动态混合重构" 的DG/FV 混合算法. 最后展望了基于非结构/混合网格的高阶精度格式的未来发展趋势及应用前景.