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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

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.

AnAdaptive Moving Mesh Method for Two-Dimensional Relativistic Hydrodynamics

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Communications in Computational Physics 2012年第1期11卷 114-146页

作者： Peng He huazhong tang HEDPS CAPT&LMAMSchool of Mathematical SciencesPeking UniversityBeijing 100871China

This paper extends the adaptive moving mesh method developed by tang and tang[36]to two-dimensional(2D)relativistic hydrodynamic(RHD)*** algorithm consists of two“independent”parts:the time evolution of the RHD equations and the(static)mesh iteration *** the first part,the RHD equations are discretized by using a high resolution finite volume scheme on the fixed but nonuniform meshes without the full characteristic decomposition of the governing *** second part is an iterative *** each iteration,the mesh points are first redistributed,and then the cell averages of the conservative variables are remapped onto the new mesh in a conservative *** numerical examples are given to demonstrate the accuracy and effectiveness of the proposed method.

关键词： Adaptive moving mesh method finite volume method conservative interpolation relativistic hydrodynamics

A High-Order Accurate Gas-Kinetic Scheme for One-and Two-Dimensional Flow Simulation

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Communications in Computational Physics 2014年第4期15卷 911-943页

作者： Na Liu huazhong tang HEDPS CAPT&LMAMSchool of Mathematical SciencesPeking UniversityBeijing 100871P.R.China

This paper develops a high-order accurate gas-kinetic scheme in the framework of the finite volume method for the one-and two-dimensional flow simulations,which is an extension of the third-order accurate gas-kinetic scheme[***,***,and ***,***.,229(2010),6715-6731]and the second-order accurate gas-kinetic scheme[***,***.,171(2001),289-335].It is formed by two parts:quartic polynomial reconstruction of the macroscopic variables and fourth-order accurate flux *** first part reconstructs a piecewise cell-center based quartic polynomial and a cell-vertex based quartic polynomial according to the“initial”cell average approximation of macroscopic variables to recover locally the non-equilibrium and equilibrium single particle velocity distribution functions around the cell *** is in view of the fact that all macroscopic variables become moments of a single particle velocity distribution function in the gas-kinetic *** generalized moment limiter is employed there to suppress the possible numerical *** the second part,the macroscopic flux at the cell interface is evolved in fourth-order accuracy by means of the simple particle transport mechanism in the microscopic level,*** transport and the Bhatnagar-Gross-Krook(BGK)*** other words,the fourth-order flux evolution is based on the solution(*** particle velocity distribution function)of the BGK model for the Boltzmann *** 1D and 2D test problems are numerically solved by using the proposed high-order accurate gas-kinetic *** comparing with the exact solutions or the numerical solutions obtained the secondorder or third-order accurate gas-kinetic scheme,the computations demonstrate that our scheme is effective and accurate for simulating invisid and viscous fluid flows,and the accuracy of the high-order GKS depends on the choice of the(numerical)collision time.

关键词： Gas-kinetic scheme initial reconstruction BGK model Boltzmann equation velocity distribution function

High-Order Accurate Entropy Stable Finite Difference Schemes for One- and Two-Dimensional Special Relativistic Hydrodynamics

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Advances in Applied Mathematics and Mechanics 2020年第1期12卷 1-29页

作者： Junming Duan huazhong tang Center for Applied Physics and Technology HEDPSand LMAMSchool of Mathematical SciencesPeking UniversityBeijing 100871China School of Mathematics and Computational Science Xiangtan UniversityXiangtanHunan 411105China

This paper develops the high-order accurate entropy stable finite difference schemes for one-and two-dimensional special relativistic hydrodynamic *** schemes are built on the entropy conservative flux and the weighted essentially non-oscillatory(WENO)technique as well as explicit Runge-Kutta time *** key is to technically construct the affordable entropy conservative flux of the semi-discrete second-order accurate entropy conservative schemes satisfying the semi-discrete entropy equality for the found convex entropy *** soon as the entropy conservative flux is derived,the dissipation term can be added to give the semidiscrete entropy stable schemes satisfying the semi-discrete entropy inequality with the given convex entropy *** WENO reconstruction for the scaled entropy variables and the high-order explicit Runge-Kutta time discretization are implemented to obtain the fully-discrete high-order entropy stable *** numerical tests are conducted to validate the accuracy and the ability to capture discontinuities of our entropy stable schemes.

关键词： Entropy conservative scheme entropy stable scheme high order accuracy finite difference scheme special relativistic hydrodynamics