基于静动态重构的高阶DG/FV混合格式在二维非结构网格中的推广

通过比较间断Galerkin有限元方法(DGM)和有限体积方法(FVM),提出"静态重构"和"动态重构"的概念,进一步建立基于静动态"混合重构"算法的三阶DG/FV混合格式.在DG/FV混合格式中,单元平均值和一阶导数由DGM方法"动态重构",二阶导数利用FVM方法"静态重构";在此基础上,构造高阶多项式插值函数,得到...     

复杂外形静动态混合网格生成技术研究新进展

网格生成技术是CFD复杂工程应用的重要环节, 网格质量的好坏直接影响计算结果的精准度, 因此其已成为CFD的重要研究领域.在张涵信院士的指导下, 作者自20世纪90年代初开始开展非结构网格、混合网格技术和相应的计算方法研究, 并逐步发展至动态混合网格技术及非定常计算方法.在张涵信院士80华诞之际, 对近年来作者及团队在网格生成技术及应用方面所取得的进展进行了简要的综述, 分别介绍了静动态混合网格生成、定常/非定常计算方法、网格技术的应用等方面的进展情况.最后, 就网格生成技术目前还存在的问题, 展望了未来的发展方向.作者谨以此文表达对张涵信院士25年来的培养、关怀和帮助的崇高敬意.      

三维拉氏理想磁流体数值模拟方法

在Z箍缩驱动ICF过程中,磁场流体耦合作用是其整个物理过程中非常重要的部分.针对Z箍缩过程中多介质、大变形的特点,发展了三维相容拉氏理想磁流体交错型以及单元中心型格式,两种格式均具有一阶时间与空间精度.通过数值算例,验证其精度和强壮性,并比较分析两种格式的特点.     

FORCE schemes on unstructured meshes I: Conservative hyperbolic systems

This paper is about the construction of numerical fluxes of the centred type for one-step schemes in conservative form for solving general systems of conservation laws in multiple space dimensions on structured and unstructured meshes. The work is a multi-dimensional extension of the one-dimensional FORCE flux and is closely related to the work of Nessyahu–Tadmor and Arminjon. The resulting basic flux is first-order accurate and monotone; it is then extended to arbitrary order of accuracy in space and time on unstructured meshes in the framework of finite volume and discontinuous Galerkin methods. The performance of the schemes is assessed on a suite of test problems for the multi-dimensional Euler and Magnetohydrodynamics equations on unstructured meshes.     

非结构网格上温度扩散方程的能流计算方法

讨论非结构网格上温度扩散方程的能流计算方法.应用有限点方法(Finite Point Method,简称FPM)导出基于有限点两点公式和三点公式的能流计算公式,该公式适用于任意多边形及非匹配网格等非结构网格;给出网格角点温度新的计算公式.数值试验表明:基于两点公式的离散解和基于三点公式的离散解均具有平方阶的收敛速度;基于三点公式的离散解的精度总优于基于两点公式的离散解.     

A multi-moment finite volume formulation for shallow water equations on unstructured mesh

A novel and accurate finite volume method has been presented to solve the shallow water equations on unstructured grid in plane geometry. In addition to the volume integrated average (VIA moment) for each mesh cell, the point values (PV moment) defined on cell boundary are also treated as the model variables. The volume integrated average is updated via a finite volume formulation, and thus is numerically conserved, while the point value is computed by a point-wise Riemann solver. The cell-wise local interpolation reconstruction is built based on both the VIA and the PV moments, which results in a scheme of almost third order accuracy. Efforts have also been made to formulate the source term of the bottom topography in a way to balance the numerical flux function to satisfy the so-called C-property. The proposed numerical model is validated by numerical tests in comparison with other methods reported in the literature.     

Efficient kinetic schemes for steady and unsteady flow simulations on unstructured meshes

This paper presents efficient second-order kinetic schemes on unstructured meshes for both compressible unsteady and incompressible steady flows. For compressible unsteady flows, a time-dependent gas distribution function with a discontinuous particle velocity space at a cell interface is constructed and used for the evaluations of both numerical fluxes and conservative flow variables. As a result, a compact scheme on the unstructured meshes is developed. For incompressible steady flows, a continuous second-order gas-kinetic BGK type scheme is presented, for which the time-dependent gas distribution function with a continuous particle velocity is used on unstructured meshes. The efficiency of the schemes lies in the fact that the slopes of the flow variables inside each cell can be constructed using values of the flow variables within that cell only without involving neighboring cells. Therefore, even with the stencil of a first-order scheme, a high resolution method is constructed. Numerical examples are presented which are compared with the benchmark solutions and the experimental measurements.     

有限点方法研究

在二维散乱离散点集上研究一类无网格方法——有限点方法(Finite Point Method,简称FPM),建立方法的基础.采用方向微商和方向差商讨论有限点方法,建立各阶各方向微商间的关系式.利用这些关系式,根据被逼近点的邻点数目差异,分别建立数值方向微商的五点公式及少点(两点、三点、四点)公式;研究五点公式的可解性条件与可允许邻点集;获得典型微分算子的数值方向微商公式等.理论分析和数值试验表明,随着邻点数目的增加,相应数值公式的逼近精度随之提高.这类近似公式不仅为在散乱离散点集上构造各类偏微分方程的格式奠定了基础,同时,也可应用于偏微分方程非结构网格计算方法,提高方法的精度.     

多重网格法在非结构网格中的应用

将多重网格法运用于非结构网格.网格是通过聚合法得到的,网格之间是相互关联的.方程的求解采用Jamson的有限体积法.给出了二维、三维情况的数值算例.     

Basic function method A new numerical method on unstructured grids

A new numerical method-basic function method is proposed. This method can directly discrete differential operators on unstructured grids. By using the expansion of basic function to approach the exact function, the central and upwind schemes of derivative are constructed. By using the polynomial as basic function, applying the technique of flux splitting method and the combination of central and upwind schemes, the non-physical fluctuation near the shock wave is suppressed. The first-order basic function scheme of polynomial type for solving inviscid compressible flow numerically is constructed in this paper. Several numerical results of many typical examples for one-, two- and three-dimensional inviscid compressible steady flow illustrate that it is a new scheme with high accuracy and high resolution for shock wave. Especially, combining with the adaptive remeshing technique, the satisfactory results can be obtained by these schemes.     

Discontinuous spectral element method for solving radiative heat transfer in multidimensional semitransparent media

A discontinuous spectral element method (DSEM) is presented to solve radiative heat transfer in multidimensional semitransparent media. This method is based on the general discontinuous Galerkin formulation. Chebyshev polynomial is used to build basis function on each element and both structured and unstructured elements are considered. The DSEM has properties such as hp-convergence, local conservation and its solutions are allowed to be discontinuous across interelement boundaries. The influences of different schemes for treatment of the interelement numerical flux on the performance of the DSEM are compared. The p-convergence characteristics of the DSEM are studied. Four various test problems are taken as examples to verify the performance of the DSEM, especially the performance to solve the problems with discontinuity in the angular distribution of radiative intensity. The predicted results by the DSEM agree well with the benchmark solutions. Numerical results show that the p-convergence rate of the DSEM follows exponential law, and the DSEM is stable, accurate and effective to solve multidimensional radiative transfer in semitransparent media.     

Local DG method using WENO type limiters for convection–diffusion problems

The local discontinuous Galerkin (LDG) method is a spatial discretization procedure for convection–diffusion equations, which employs useful features from high resolution finite volume schemes, such as the exact or approximate Riemann solvers serving as numerical fluxes and limiters, which is termed as Runge–Kutta LDG (RKLDG) when TVD Runge–Kutta method is applied for time discretization. It has the advantage of flexibility in handling complicated geometry, h-p adaptivity, and efficiency of parallel implementation and has been used successfully in many applications. However, the limiters used to control spurious oscillations in the presence of strong shocks are less robust than the strategies of essentially non-oscillatory (ENO) and weighted ENO (WENO) finite volume and finite difference methods. In this paper, we investigated RKLDG methods with WENO and Hermite WENO (HWENO) limiters for solving convection–diffusion equations on unstructured meshes, with the goal of obtaining a robust and high order limiting procedure to simultaneously obtain uniform high order accuracy and sharp, non-oscillatory shock transition. Numerical results are provided to illustrate the behavior of these procedures.     

L-stability of vertex-based MUSCL finite volume schemes on unstructured grids: Simulation of incompressible flows with high density ratios

This work is devoted to the design of multi-dimensional finite volume schemes for solving transport equations on unstructured grids. In the framework of MUSCL vertex-based methods we construct numerical fluxes such that the local maximum property is guaranteed under an explicit Courant–Friedrichs–Levy condition. The method can be naturally completed by adaptive local mesh refinements and it turns out that the mesh generation is less constrained than when using the competitive cell-centered methods. We illustrate the effectiveness of the scheme by simulating variable density incompressible viscous flows. Numerical simulations underline the theoretical predictions and succeed in the computation of high density ratio phenomena such as a water bubble falling in air.     

On the characteristics-based ACM for incompressible flows

In this paper, the revised characteristics-based (CB) method for incompressible flows recently derived by Neofytou [P. Neofytou, Revision of the characteristic-based scheme for incompressible flows, J. Comput. Phys. 222 (2007) 475–484] has been further investigated. We have derived all the formulas for pressure and velocities from this revised CB method, which is based on the artificial compressibility method (ACM) [A.J. Chorin, A numerical solution for solving incompressible viscous flow problems, J. Comput. Phys. 2 (1967) 12]. Then we analyze the formulations of the original CB method [D. Drikakis, P.A. Govatsos, D.E. Papatonis, A characteristic based method for incompressible flows, Int. J. Numer. Meth. Fluids 19 (1994) 667–685; E. Shapiro, D. Drikakis, Non-conservative and conservative formulations of characteristics numerical reconstructions for incompressible flows, Int. J. Numer. Meth. Eng. 66 (2006) 1466–1482; D. Drikakis, P.K. Smolarkiewicz, On spurious vortical structures, J. Comput. Phys. 172 (2001) 309–325; F. Mallinger, D. Drikakis, Instability in three-dimensional, unsteady stenotic flows, Int. J. Heat Fluid Flow 23 (2002) 657–663; E. Shapiro, D. Drikakis, Artificial compressibility, characteristics-based schemes for variable density, incompressible, multi-species flows. Parts I. Derivation of different formulations and constant density limit, J. Comput. Phys. 210 (2005) 584–607; Y. Zhao, B. Zhang, A high-order characteristics upwind FV method for incompressible flow and heat transfer simulation on unstructured grids, Comput. Meth. Appl. Mech. Eng. 190 (5–7) (2000) 733–756] to investigate their consistency with the governing flow equations after convergence has been achieved. Furthermore we have implemented both formulations in an unstructured-grid finite volume solver [Y. Zhao, B. Zhang, A high-order characteristics upwind FV method for incompressible flow and heat transfer simulation on unstructured grids, Comput. Meth. Appl. Mech. Eng. 190 (5–7) (2000) 733–756]. Detailed numerical experiments show that both methods give almost identical solutions and convergence rates. Both can generate solutions which agree well with published results and experimental measurements. We thus conclude that both methods, being upwind schemes designed for the ACM, have the same performances in terms of accuracy and convergence speed, even though the revised method is more complex with less stringent assumptions made, while the original CB method is simpler due to the use of extra simplifying assumptions.     

A first-order system approach for diffusion equation. I: Second-order residual-distribution schemes

In this paper, we embark on a new strategy for computing the steady state solution of the diffusion equation. The new strategy is to solve an equivalent first-order hyperbolic system instead of the second-order diffusion equation, introducing solution gradients as additional unknowns. We show that schemes developed for the first-order system allow O(h) time step instead of O(h²) and converge very rapidly toward the steady state. Moreover, this extremely fast convergence comes with the solution gradients (viscous stresses/heat fluxes for the Navier–Stokes equations) simultaneously computed with the same order of accuracy as the main variable. The proposed schemes are formulated as residual-distribution schemes (but can also be identified as finite-volume schemes), directly on unstructured grids. We present numerical results to demonstrate the tremendous gains offered by the new diffusion schemes, driving the rise of explicit schemes in the steady state computation for diffusion problems.     

The multi-dimensional limiters for solving hyperbolic conservation laws on unstructured grids II: Extension to high order finite volume schemes

In this paper, the multidimensional limiter for the second order finite volume schemes on the unstructured grid, namely the Weighted Biased Average procedure developed in our previous paper is extended to high order finite volume schemes solving hyperbolic conservation laws. This extension relies on two key techniques: the secondary reconstruction and the successive limiting procedure. These techniques are discussed in detail in the present paper. Numerical experiments shows that this limiting procedure is very effective in removing numerical oscillations in the vicinity of discontinuities. And furthermore this procedure is efficient, robust and accuracy preserving.     

Explicit and implicit finite difference schemes for fractional Cattaneo equation

In this paper, the numerical solution of fractional (non-integer)-order Cattaneo equation for describing anomalous diffusion has been investigated. Two finite difference schemes namely an explicit predictor–corrector and totally implicit schemes have been developed. In developing each scheme, a separate formulation approach for the governing equations has been considered. The explicit predictor–corrector scheme is the fractional generalization of well-known MacCormack scheme and has been called Generalized MacCormack scheme. This scheme solves two coupled low-order equations and simultaneously computes the flux term with the main variable. Fully implicit scheme however solves a single high-order undecomposed equation. For Generalized MacCormack scheme, stability analysis has been studied through Fourier method. Through a numerical test, the experimental order of convergency of both schemes has been found. Then, the domain of applicability and some numerical properties of each scheme have been discussed.     

自适应间断有限元方法求解三维欧拉方程

将龙格库塔间断有限元方法(RDDG)与自适应方法相结合,求解三维欧拉方程.区域剖分采用非结构四面体网格,依据数值解的变化采用自适应技术对网格进行局部加密或粗化,减少总体网格数目,提高计算效率.给出四种自适应策略并分析不同自适应策略的优缺点.数值算例表明方法的有效性.     

A class of hybrid DG/FV methods for conservation laws I: Basic formulation and one-dimensional systems

By comparing the discontinuous Galerkin (DG) and the finite volume (FV) methods, a concept of ‘static reconstruction’ and ‘dynamic reconstruction’ is introduced for high-order numerical methods. Based on the new concept, a class of hybrid DG/FV schemes is presented for one-dimensional conservation law using a ‘hybrid reconstruction’ approach. In the hybrid DG/FV schemes, the lower-order derivatives of a piecewise polynomial solution are computed locally in a cell by the DG method based on Taylor basis functions (called as ‘dynamic reconstruction’), while the higher-order derivatives are re-constructed by the ‘static reconstruction’ of the FV method, using the known lower-order derivatives in the cell itself and its adjacent neighboring cells. The hybrid DG/FV methods can greatly reduce CPU time and memory required by the traditional DG methods with the same order of accuracy on the same mesh, and they can be extended directly to unstructured and hybrid grids in two and three dimensions similar to the DG and/or FV methods. The hybrid DG/FV methods are applied to one-dimensional conservation law, including linear and non-linear scalar equation and Euler equations. In order to capture the strong shock waves without spurious oscillations, a simple shock detection approach is developed to mark ‘trouble cells’, and a moment limiter is adopted for higher-order schemes. The numerical results demonstrate the accuracy, and the super-convergence property is shown for the third-order hybrid DG/FV schemes. In addition, by analyzing the eigenvalues of the semi-discretized system in one dimension, we discuss the spectral properties of the hybrid DG/FV schemes to explain the super-convergence phenomenon.