汤华中
从事微分方程数值解及流体力学中数值计算方面的研究工作、自适应网格方法、科学与工程计算等方面的研究
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 姓名：汤华中
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学术头衔：
博士生导师， 教育部“新世纪优秀人才支持计划”入选者， 国家杰出青年科学基金获得者
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学科领域：
计算数学
 研究兴趣：从事微分方程数值解及流体力学中数值计算方面的研究工作、自适应网格方法、科学与工程计算等方面的研究
汤华中，北京大学数学科学学院教授，博士生导师，主要从事微分方程数值解及流体力学中数值计算方面的研究工作、自适应网格方法、科学与工程计算等方面的研究。现任科学与工程计算系系主任(2008.9)，北京市计算数学学会第八届理事长(2009.12)，中国计算数学学会常务理事、副秘书长(2006.12), 中国工业与应用数学学会副秘书长(2008.10)，“Journal of Computational Physics”， “International Journal for Numerical Methods in Fluids”和“计算物理”等国内外杂志编委，发表论文50余篇，出版教材《微分方程数值解法》（合编）一部。1997年与合作者获航空航天工业部科技进步奖二等奖1项，2001年获得德国洪堡基金会研究奖学金(Research fellow of the Alexander von Humboldt foundation)， 2007年入选教育部“新世纪优秀人才支持计划”, 2007年与合作者获高校科学技术奖(自然科学一等) 1项；2009年获国家杰出青年基金。
个人主页：http://dsec.pku.edu.cn/~tanghz

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14
【期刊论文】An adaptive moving mesh method for twodimensional ideal magnetohydrodynamics
汤华中， Jianqiang Han， Huazhong Tang *
Journal of Computational Physics 220(2007)791812，0001，（）：
1年11月30日
This paper presents an adaptive moving mesh algorithm for twodimensional (2D) ideal magnetohydrodynamics (MHD) that utilizes a staggered constrained transport technique to keep the magnetic field divergencefree. The algorithm consists of two independent parts: MHD evolution and meshredistribution. The first part is a highresolution, divergencefree, shockcapturing scheme on a fixed quadrangular mesh, while the second part is an iterative procedure. In each iteration, mesh points are first redistributed, and then a conservativeinterpolation formula is used to calculate the remapped cellaverages of the mass, momentum, and total energy on the resulting new mesh; the magnetic potential is remapped to the new mesh in a nonconservative way and is reconstructed to give a divergencefree magnetic field on the new mesh. Several numerical examples are given to demonstrate that the proposed method can achieve high numerical accuracy, track and resolve strong shock waves in ideal MHD problems, and preserve divergencefree property of the magnetic field. Numerical examples include the smooth Alfve´n wave problem, 2D and 2.5D shock tube problems, two rotor problems, the stringent blast problem, and the cloudshock interaction problem.
Adaptive moving mesh method， Finite volume method， Constrained transport， Magnetohydrodynamics， Divergencefree

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【期刊论文】An adaptive GRP scheme for compressible fluid flows
汤华中， Ee Han and Jiequan Li Huazhong Tang
，0001，（）：
1年11月30日
This paper presents a secondorder accurate adaptive generalized Riemann problem (GRP) scheme for one and two dimensional compressible fluid flows. The current scheme consists of two independent parts: Mesh redistribution and PDE evolution. The first part is an iterative procedure. In each iteration, mesh points are first redistributed, and then a conservativeinterpolation formula is used to calculate the cellaverages and the slopes of conservative variables on the resulting new mesh. The second part is to evolve the compressible fluid flows on a fixed nonuniform mesh with the Eulerian GRP scheme, which is directly extended to two dimensional arbitrary quadrilateral meshes. Several numerical examples show that the current adaptive GRP scheme does not only improve the resolution as well as accuracy of numerical solutions with a few mesh points, but also reduces possible errors or oscillations effectively.
GRP scheme,， adaptive moving mesh method,， monitor function,， conservative interpolation.，

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【期刊论文】LOCAL OSCILLATIONS IN FINITE DIFFERENCE SOLUTIONS OF HYPERBOLIC CONSERVATION LAWS
汤华中， JIEQUAN LI， HUAZHONG TANG， GERALD WARNECKE AND LUMEI ZHANG
，0001，（）：
1年11月30日
It was generally expected that monotone schemes are oscillationfree for hyperbolic conservation laws. However, recently local oscillations were observed and usually understood to be caused by relative phase errors. In order to further explain this, we first investigate the discretization of initial data that trigger the chequerboard mode, the highest frequency mode. Then we proceed to use the discrete Fourier analysis and the modified equation analysis to distinguish the dissipative and dispersive effects of numerical schemes for low frequency and high frequency modes, respectively. It is shown that the relative phase error is of order O(1) for the high frequency modes unj=λnkeiξj, ξ≈π, but of order O(ξ2) for low frequency modes (ξ≈0). In order to avoid numerical oscillations, the relative phase errors should be offset by numerical dissipation of at least the same order. Numerical damping, i.e. the zero order term in the corresponding modified equation, is important to dissipate the oscillations caused by the relative phase errors of high frequency modes. This is in contrast to the role of numerical viscosity, the second order term, which is the lowest order term usually present to suppress the relative phase errors of low frequency modes.
Finite difference schemes,， high and low frequency modes,， oscillations,， chequerboard modes,， numerical damping,， numerical viscosity,， relative phase error,， modifiedequation analysis,， discrete Fourier analysis.，

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【期刊论文】An Adaptive Ghost Fluid Finite Volume Method for Compressible GasWater Simulations
汤华中， Chunwu Wang Huazhong Tang★ Tiegang Liu
，0001，（）：
1年11月30日
An adaptive ghost fluidnite volume method is developed for oneand twodimensional compressible multimedium flows in this work. It couples the real ghost fluid method (GFM) [SIAM J. Sci. Comput. 28 (2006) 278] and the adaptive moving mesh method [SIAM J. Numer. Anal. 41(2003) 487; J. Comput. Phys. 188(2003) 543], and thus retains their advantages. This work shows that the local mesh clustering in the vicinity of the material interface can effectively reduce both numerical and conservative errors caused by the GFM around the material interface and other discontinuities. Besides the improvement of flow field resolution, the adaptive ghost fluid method also largely increases the computational efficiency. Several numerical experiments are conducted to demonstrate robustness and efficiency of the current method. They include several 1D and 2D gaswater flow problems, involving a large density gradient at the material interface and strong shockinterface interactions. The results show that our algorithm can capture the shock waves and the material interface accurately, and is stable and robust even solution with large density and pressure gradients.
Finite volume method， ghost fluid method， moving mesh method， levelset method， approximate Riemann solver， gaswater Riemann problem

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【期刊论文】SecondOrder Accurate Godunov Scheme for Multicomponent Flows on Moving Triangular Meshes
汤华中， Guoxian Chen • Huazhong Tang • Pingwen Zhang
J Sci Comput (2008) 34: 6486，0001，（）：
1年11月30日
This paper presents a secondorder accurate adaptive Godunov method for twodimensional (2D) compressible multicomponent flows, which is an extension of the previous adaptive moving mesh method of Tang et al. (SIAM J. Numer. Anal. 41: 487515, 2003) to unstructured triangular meshes in place of the structured quadrangular meshes. The current algorithm solves the governing equations of 2D multicomponent flows and the finitevolume approximations of the mesh equations by a fully conservative, secondorder accurate Godunov scheme and a relaxed Jacobitype iteration, respectively. The geometrybased conservative interpolation is employed to remap the solutions from the old mesh to the newly resulting mesh, and a simple slope limiter and a new monitor function are chosen to obtain oscillationfree solutions, and track and resolve both small, local, and large solution gradients automatically. Several numerical experiments are conducted to demonstrate robustness and efficiency of the proposed method. They are a quasi2D Riemann problem, the doubleMach reflection problem, the forward facing step problem, and two shock wave and bubble interaction problems.
Adaptive moving mesh method • Finite volume method • Godunov scheme • Multicomponent flows • Unstructured mesh

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【期刊论文】An adaptive phase field method for the mixture of two incompressible fluids
汤华中， Zhengru Zhang a， Huazhong Tang b， *
Computers & Fluids 36(2007)13071318，0001，（）：
1年11月30日
This paper develops an adaptive moving mesh method to solve a phase field model for the mixture of two incompressible fluids. The projection method is implemented on a halfstaggered, moving quadrilateral mesh to keep the velocity field divergencefree, and the conjugate gradient or multigrid method is employed to solve the discrete Poisson equations. The current algorithm is composed by two independent parts: evolution of the governing equations and meshredistribution. In the first part, the incompressible NavierStokes equations are solved on a fixed halfstaggered mesh by the rotational incremental pressurecorrection scheme, and the AllenCahn type of phase equation is approximated by a conservative, secondorder accurate central difference scheme, where the Lagrangian multiplier is used to preserve the massconservation of the phase field. The second part is an iteration procedure. During the mesh redistribution, the phase field is remapped onto the newly resulting meshes by the highresolution conservative interpolation, while the nonconservative interpolation algorithm is applied to the velocity field. The projection technique is used to obtain a divergencefree velocity field at the end of this part. The resultant numerical scheme is stable, mass conservative, highly efficient and fast, and capable of handling variable density and viscosity. Several numerical experiments are presented to demonstrate the efficiency and robustness of the proposed algorithm.

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【期刊论文】An efficient adaptive mesh redistribution method for a nonlinear Dirac equation
汤华中， Han Wang， Huazhong Tang *
Journal of Computational Physics 222(2007)176193，0001，（）：
1年11月30日
This paper presents an efficient adaptive mesh redistribution method to solve a nonlinear Dirac (NLD) equation. Our algorithm is formed by three parts: the NLD evolution, the iterative mesh redistribution of the coarse mesh and the local uniform refinement of the final coarse mesh. At each time level, the equidistribution principle is first employed to iteratively redistribute coarse mesh points, and the scalar monitor function is subsequently interpolated on the coarse mesh in order to do one new iteration and improve the grid adaptivity. After an adaptive coarse mesh is generated ideally and finally, each coarse mesh interval is equally divided into some fine cells to give an adaptive fine mesh of the physical domain, and then the solution vector is remapped on the resulting new fine mesh by an affine method. The NLD equation is finally solved by using a high resolution shockcapturing method on the (fixed) nonuniform fine mesh. Extensive numerical experiments demonstrate that the proposed adaptive mesh method gives the thirdorder rate of convergence, and yields an efficient and fast NLD solver that tracks and resolves both small, local and large solution gradients automatically.
Adaptive mesh redistribution， The Dirac equation， Local uniform refinement， Solitary wave， High resolution scheme

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【期刊论文】Short note A note on the conservative schemes for the Euler equations
汤华中， Huazhong Tang a， *， Tiegang Liu b
Journal of Computational Physics 218(2006)451459，0001，（）：
1年11月30日
This note gives a numerical investigation for the popular high resolution conservative schemes when applied to inviscid, compressible, perfect gas flows with an initial high density ratio as well as a high pressure ratio. The results show that they work very inefficiently and may give inaccurate numerical results even over a very fine mesh when applied to such a problem. Numerical tests show that increasing the order of accuracy of the numerical schemes does not help much in improving the numerical results. How to cure this difficulty is still open.
High resolution schemes， Godunov scheme， The Euler equations， Rarefaction wave， Shock wave

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【期刊论文】Interaction for the solitary waves of a nonlinear Dirac model
汤华中， Sihong Shao， Huazhong Tang ∗
Physics Letters A 345(2005)119128，0001，（）：
1年11月30日
This Letter presents a numerical study of the interaction dynamics for the solitary waves of a nonlinear Dirac field with scalar selfinteraction by using a fourth order accurate RungeKutta discontinuous Galerkin (RKDG) method. Some new interaction phenomena are observed: (a) a new quasistable longlived oscillating bound state from the binary collisions of a singlehumped soliton and a twohumped soliton; (b) collapse in binary and ternary collisions; (c) strongly inelastic interaction in ternary collisions; and (d) bound states with a short or long lifetime from ternary collisions.
RungeKutta discontinuous Galerkin method， Dirac model， Bound state， Interaction dynamics

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汤华中， HUAZHONG TANG† AND GERALD WARNECKE‡
SIAM J, SCI, COMPUT Vol.0, No.0, pp. 000000，0001，（）：
1年11月30日
Based on a simple projection of the solution increments of the underlying partial differential equations (PDEs) at each local time level, this paper presents a difference scheme for nonlinear HamiltonJacobi (HJ) equations with varying time and space grids. The scheme is of good consistency and monotone under a local CFLtype condition. Moreover, one may deduce a conservative local time step scheme similar to Osher and Sanders scheme approximating hyperbolic conservation law (CL) from our scheme according to the close relation between CLs and HJ equations. Second order accurate schemes are constructed by combining the reconstruction technique with a second order accurate RungeKutta time discretization scheme or a LaxWendroff type method. They keep some good properties of the global time step schemes, including stability and convergence, and can be applied to solve numerically the initialboundaryvalue problems of viscous HJ equations. They are also suitable to parallel computing. Numerical errors and the experimental rate of convergence in the Lpnorm, p=1, 2, and ∞, are obtained for several oneand twodimensional problems. The results show that the present schemes are of higher order accuracy.
HamiltonJacobi equation,， finite difference scheme,， local time step discretization,， NavierStokes equations

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