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Numerical Mathematics(Theory,Methods and Applications) 2010年第4期3卷 387-404页

作者： huazhong CAPT & LMAM of School of Mathematical Sciences Peking University Beijing 100871 China Department of Mathematics The Hong Kong University of Science and Technology Clear Water Bay Kowloon Hong Kong Department of Mechanical & Aerospace Engineering New Mexico State University Las Cruces NM 88003-8001 USA

The gas-kinetic theory based flux splitting method has been successfully proposed for solving one-and two-dimensional ideal magnetohydrodynamics by Xu et al. [***.,1999;2000],*** paper extends the kinetic method to solve three-dimensional ideal magnetohydrodynamics equations,where an adaptive parameter 17 is used to control the numerical dissipation in the flux splitting method. Several numerical examples are given to demonstrate that the proposed method can achieve high numerical accuracy and resolve strong discontinuous waves in three dimensional ideal MHD problems.

关键词： The kinetic BGK scheme magnetohydrodynamics divergence-free condition

A Newton multigrid method for steady-state shallow water equations with topography and dry areas

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Applied Mathematics and Mechanics(English Edition) 2016年第11期37卷 1441-1466页

作者： Kailiang WU huazhong TANG LMAM School of Mathematical Sciences Peking University HEDPS CAPT & LMAM School of Mathematical SciencesPeking University School of Mathematics and Computational Science Xiangtan University

A Newton multigrid method is developed for one-dimensional （1D） and two- dimensional （2D） steady-state shallow water equations （SWEs） with topography and dry areas. The nonlinear system arising from the well-balanced finite volume discretization of the steady-state SWEs is solved by the Newton method as the outer iteration and a geometric multigrid method with the block symmetric Gauss-Seidel smoother as the inner iteration. The proposed Newton multigrid method makes use of the local residual to regularize the Jacobian matrix of the Newton iteration, and can handle the steady- state problem with wet/dry transition. Several numerical experiments are conducted to demonstrate the efficiency, robustness, and well-balanced property of the proposed method. The relation between the convergence behavior of the Newton multigrid method and the distribution of the eigenvalues of the iteration matrix is detailedly discussed.

关键词： Newton method multigrid block symmetric Gauss-Seidel shallow waterequation (SWE) steady-state solution

AN EFFICIENT ADER DISCONTINUOUS GALERKIN SCHEME FOR DIRECTLY SOLVING HAMILTON-JACOBI EQUATION

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Journal of Computational Mathematics 2020年第1期38卷 58-83页

作者： Junming Duan huazhong Tang HEDPS CAPT&LMAMSchool of Mathematical SciencesPeking UniversityBeijing 100871China School of Mathematics and Computational Science Xiangtan UniversityXiangtan 411105China

This paper proposes an efficient ADER(Arbitrary DERivatives in space and time)discontinuous Galerkin(DG)scheme to directly solve the Hamilton-Jacobi *** multi-stage Runge-Kutta methods used in the Runge-Kutta DG(RKDG)schemes,the ADER scheme is one-stage in time discretization,which is desirable in many *** ADER scheme used here relies on a local continuous spacetime Galerkin predictor instead of the usual Cauchy-Kovalewski procedure to achieve high order accuracy both in space and *** such predictor step,a local Cauchy problem in each cell is solved based on a weak formulation of the original equations in *** resulting spacetime representation of the numerical solution provides the temporal accuracy that matches the spatial accuracy of the underlying DG *** scheme is formulated in the modal space and the volume integral and the numerical fluxes at the cell interfaces can be explicitly *** explicit formulae of the scheme at third order is provided on two-dimensional structured *** computational complexity of the ADER-DG scheme is compared to that of the RKDG *** experiments are also provided to demonstrate the accuracy and efficiency of our scheme.

关键词： Hamilton-Jacobi equation ADER Discontinuous Galerkin methods Local continuous spacetime Galerkin predictor High order accuracy

TWO-STAGE FOURTH-ORDER ACCURATE TIME DISCRETIZATIONS FOR 1D AND 2D SPECIAL RELATIVISTIC HYDRODYNAMICS

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Journal of Computational Mathematics 2020年第5期38卷 768-796页

作者： Yuhuan Yuan huazhong Tang Center for Applied Physics and Technology HEDPSand LMAMSchool of Mathematical SciencesPeking UniversityBeijing 100871China

This paper studies the two-stage fourth-order accurate time discretization[*** and ***,SIAM ***.,38(2016)]and its application to the special relativistic hydrodynamical *** analysis reveals that the new two-stage fourth-order accurate time discretizations can be *** the aid of the direct Eulerian GRP(generalized Riemann problem)methods and the analytical resolution of the local“quasi 1D”GRP,the two-stage fourth-order accurate time discretizations are successfully implemented for the 1D and 2D special relativistic hydrodynamical *** numerical experiments demonstrate the performance and accuracy as well as robustness of our schemes.

关键词： Time discretization Shock-capturing scheme GRP method Relativistic hydrodynamics Hyperbolic conservation laws

Globally hyperbolic moment model of arbitrary order for the three-dimensional special relativistic Boltzmann equation with the Anderson-Witting collision

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Science China Mathematics 2022年第5期65卷 1029-1064页

作者： Yangyu Kuang huazhong Tang Center for Applied Physics and Technology HEDPS and LMAMSchool of Mathematical SciencesPeking UniversityBeijing 100871China

This paper continues to derive the globally hyperbolic moment model of arbitrary order for the three-dimensional special relativistic Boltzmann equation with the Anderson-Witting *** method is the model reduction by the operator *** an orthogonal basis of the weighted polynomial space is crucial and built on infinite families of the complicate relativistic Grad type orthogonal polynomials depending on a parameter and the real spherical harmonics instead of the irreducible *** study the properties of those functions carefully,including their recurrence relations,their derivatives with respect to the independent variable and the parameter,and the zeros of the orthogonal *** moment model is proved to be globally hyperbolic and linearly ***,the Lorentz covariance,the quasi-one-dimensional case,and the non-relativistic and ultra-relativistic limits are also studied.

关键词： moment method hyperbolicity special relativistic Boltzmann equation model reduction operator projection

ON THE EXPLICIT TWO-STAGE FOURTH-ORDER ACCURATE TIME DISCRETIZATIONS

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Journal of Computational Mathematics 2023年第2期41卷 305-324页

作者： Yuhuan Yuan huazhong Tang Center for Applied Physics and Technology HEDPS and LMAMSchool of Mathematical SciencesPeking UniversityBeijing 100871China Nanchang Hangkong University Nanchang 330063China

This paper continues to study the explicit two-stage fourth-order accurate time discretizations[5-7].By introducing variable weights,we propose a class of more general explicit one-step two-stage time discretizations,which are different from the existing methods,*** Euler methods,Runge-Kutta methods,and multistage multiderivative methods *** study the absolute stability,the stability interval,and the intersection between the imaginary axis and the absolute stability *** results show that our two-stage time discretizations can be fourth-order accurate conditionally,the absolute stability region of the proposed methods with some special choices of the variable weights can be larger than that of the classical explicit fourth-or fifth-order Runge-Kutta method,and the interval of absolute stability can be almost twice as much as the *** numerical experiments are carried out to demonstrate the performance and accuracy as well as the stability of our proposed methods.

关键词： Multistage multiderivative methods Runge-Kutta methods Absolute stability region Interval of absolute stability

Data-Domain Wave Equation Reflection Traveltime Tomography

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Journal of Earth Science 2015年第4期26卷 487-494页

作者： Bo Feng huazhong Wang Wave Phenomena and Inversion Imaging Group School of Ocean and Earth ScienceTongji University

Estimation of an accurate macro velocity model plays an important role in seismic imag- ing and model parameter inversion. Full waveform inversion （FWI） is the classical data-domain inver- sion method. However, the misfit function of FWI is highly nonlinear, and the local optimization cannot prevent convergence of the misfit function toward local minima. To converge to the global minimum, FWI needs a good initial model or reliable low frequency component and long offset data. In this article, we present a wave-equation-based reflection traveltime tomography （WERTT） method, which can pro- vide a good background model （initial model） for FWI and （least-square） pre-stack depth migration （LS-PSDM）. First, the velocity model is decomposed into a low-wavenumber component （background velocity） and a high-wavenumber component （reflectivity）. Second, the primary reflection wave is pre- dicted by wave-equation demigration, and the reflection traveltime is calculated by an automatic picking method. Finally, the misfit function of the 12-norm of the reflection traveltime residuals is mini- mized by a gradient-based method. Numerical tests show that the proposed method can invert a good background model, which can be used as an initial model for LS-PSDM or FWI.

关键词： reflection traveltime tomography primary reflection wave wave equation demigration,background velocity building rough reflectivity.

High-Order Accurate Runge-Kutta (Local) Discontinuous Galerkin Methods for One- and Two-Dimensional Fractional Diffusion Equations

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Numerical Mathematics(Theory,Methods and Applications) 2012年第3期5卷 333-358页

作者： Xia Ji huazhong Tang LSEC Institute of Computational MathematicsChinese Academy of SciencesP.O.Box 2719Beijing 100190China HEDPS CAPT and LMAMSchool of Mathematical SciencesPeking UniversityBeijing 100871China

As the generalization of the integer order partial differential equations(PDE),the fractional order PDEs are drawing more and more attention for their applications in fluid flow,finance and other *** paper presents high-order accurate Runge-Kutta local discontinuous Galerkin(DG)methods for one-and two-dimensional fractional diffusion equations containing derivatives of fractional order in *** Caputo derivative is chosen as the representation of spatial derivative,because it may represent the fractional derivative by an integral *** numerical examples show that the convergence orders of the proposed local Pk–DG methods are O(hk+1)both in one and two dimensions,where Pk denotes the space of the real-valued polynomials with degree at most k.

关键词： Discontinuous Galerkin method Runge-Kutta time discretization fractional derivative Caputo derivative diffusion equation

Interaction of Solitary Waves with a Phase Shift in a Nonlinear Dirac Model

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Communications in Computational Physics 2008年第4期3卷 950-967页

作者： Sihong Shao huazhong Tang LMAM and School of Mathematical Sciences Peking UniversityBeijing 100871China.

This paper presents a further numerical study of the interaction dynamics for solitary waves in a nonlinear Dirac model with scalar self-interaction,the Soler model,by using a fourth order accurate Runge-Kutta discontinuous Galerkin *** phase plane method is employed for the first time to analyze the interaction of Dirac solitary waves and reveals that the relative phase of those waves may vary with the *** general,the interaction of Dirac solitary waves depends on the initial phase *** two equal solitary waves are in-phase or out-of-phase initially,so are they during the interaction;if the initial phase shift is far away from 0 andπ,the relative phase begins to periodically evolve after a finite *** the interaction of out-of-phase Dirac solitary waves,we can observe:(a)full repulsion in binary and ternary collisions,depending on the distance between initial waves;(b)repulsing first,attracting afterwards,and then collapse in binary and ternary collisions of initially resting two-humped waves;(c)one-overlap interaction and two-overlap interaction in ternary collisions of initially resting waves.

关键词： Discontinuous Galerkin method phase plane method Dirac field Soler model solitary waves phase shift.