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A second order finite difference-spectral method for space fractional diffusion equations

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Science China Mathematics 2014年第6期57卷 1303-1317页

作者： HUANG JianFei NIE NingMing tang yifa LSEC ICMSECAcademy of Mathematics and Systems ScienceChinese Academy of Sciences Supercomputing Center Computer Network Information CenterChinese Academy of Sciences

A high order finite difference-spectral method is derived for solving space fractional diffusion equations,by combining the second order finite difference method in time and the spectral Galerkin method in *** stability and error estimates of the temporal semidiscrete scheme are rigorously discussed,and the convergence order of the proposed method is proved to be O(τ2+Nα-m)in L2-norm,whereτ,N,αand m are the time step size,polynomial degree,fractional derivative index and regularity of the exact solution,*** experiments are carried out to demonstrate the theoretical analysis.

关键词： space fractional diffusion equation Crank-Nicolson scheme spectral method stability conver-gence

Convergence Analysis of a Block-by-Block Method for Fractional Differential Equations

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Numerical Mathematics(Theory,Methods and Applications) 2012年第2期5卷 229-241页

作者： Jianfei Huang yifa tang Luis Vázquez LSEC ICMSECAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China Departamento deMatemática Aplicada Facultad de InformáticaInstituto deMatemática Interdisciplinar(IMI)Universidad Complutense de Madrid28040-MadridSpain

The block-by-block method,proposed by Linz for a kind of Volterra integral equations with nonsingular kernels,and extended by Kumar and Agrawal to a class of initial value problems of fractional differential equations(FDEs)with Caputo derivatives,is an efficient and stable *** analytically prove and numerically verify that this method is convergent with order at least 3 for any fractional order indexα>0.

关键词： Fractional differential equation Caputo derivative block-by-block method convergence analysis

Finite element multigrid method for multi-term time fractional advection diffusion equations

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International Journal of Modeling, Simulation, and Scientific Computing 2015年第1期6卷 1-25页

作者： Weiping Bu Xiangtao Liu yifa tang Jiye Yang LSEC ICMSECAcademy of Mathematics and Systems Science Chinese Academy of SciencesBeijing 100190P.R.China Cipher Ground 675 Rt 1 South North BrunswickNJ 08902USA School of Mathematics and Computer Ningxia University Yinchuan 750021P.R.China

In this paper,a class of multi-term time fractional advection diffusion equations(MTFADEs)is *** finite difference method in temporal direction and finite element method in spatial direction,two fully discrete schemes of MTFADEs with different definitions on multi-term time fractional derivative are *** stability and convergence of these numerical schemes are ***,a V-cycle multigrid method is proposed to solve the resulting linear *** convergence of the multigrid method is ***,some numerical examples are given for verification of our theoretical analysis.

关键词： Multi-term time fractional advection diffusion equation finite element method stability convergence V-cycle multigrid method

Convergence of Physics-Informed Neural Networks Applied to Linear Second-Order Elliptic Interface Problems

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Communications in Computational Physics 2023年第2期33卷 596-627页

作者： Sidi Wu Aiqing Zhu yifa tang Benzhuo Lu State Key Laboratory of Scientific and Engineering Computing National Center for Mathematics and Interdisciplinary SciencesAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China.

With the remarkable empirical success of neural networks across diverse scientific disciplines,rigorous error and convergence analysis are also being developed and ***,there has been little theoretical work focusing on neural networks in solving interface *** this paper,we perform a convergence analysis of physics-informed neural networks(PINNs)for solving second-order elliptic interface ***,we consider PINNs with domain decomposition technologies and introduce gradient-enhanced strategies on the interfaces to deal with boundary and interface jump *** is shown that the neural network sequence obtained by minimizing a Lipschitz regularized loss function converges to the unique solution to the interface problem in H2 as the number of samples *** experiments are provided to demonstrate our theoretical analysis.

关键词： Elliptic interface problems generalization errors convergence analysis neural networks.

ALTERNATING DIRECTION IMPLICIT SCHEMES FOR THE TWO-DIMENSIONAL TIME FRACTIONAL NONLINEAR SUPER-DIFFUSION EQUATIONS

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Journal of Computational Mathematics 2019年第3期37卷 297-315页

作者： Jianfei Huang Yue Zhao Sadia Arshad Kuangying Li yifa tang College of Mathematical Sciences Yangzhou UniversityYangzhou 225002China LSEC ICMSECAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China COMSATS Institute of Information Technology LahoTePakistan McDougall School of Petroleum Engineering The University of TulsaTulsaOK 74104USA

As is known,there exist numerous alternating direction implicit(ADI)schemes for the two-dimensional linear time fractional partial differential equations(PDEs).However,if the ADI schemes for linear problems combined with local linearization techniques are applied to solve nonlinear problems,the stability and convergence of the methods are often not *** this paper,two ADI schemes are developed for solving the two-dimensional time fractional nonlinear super-diffusion equations based on their equivalent partial integrodifferential *** these two schemes,the standard second-order central difference approximation is used for the spatial discretization,and the classical first-order approximation is applied to discretize the Riemann-Liouville fractional integral in *** solvability,unconditional stability and L2 norm convergence of the proposed ADI schemes are proved *** convergence order of the schemes is 0(τ+hx^2+hy^2),where τ is the temporal mesh size,hx and hy are spatial mesh sizes in the x and y directions,***,numerical experiments are carried out to support the theoretical results and demonstrate the performances of two ADI schemes.

关键词： Time fractional super-diffusion equation Nonlinear system ADI schemes Stability Convergence

NON-EXISTENCE OF CONJUGATE-SYMPLECTIC MULTI-STEP METHODS OF ODD ORDER

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Journal of Computational Mathematics 2007年第6期25卷 690-696页

作者： Yandong Jiao Guidong Dai Quandong Feng yifa tang LSEC ICMSEC Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijin9 100080 China Graduate School of the Chinese Academy of Sciences Beijin9 100080 China LSEC ICMSEC Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100080 China

We prove that any linear multi-step method G1^T of the form ∑k=0^mαkZk = T∑k=0^mβkJ^-1↓ΔH（Zk） with odd order u （u≥ 3） cannot be conjugate to a symplectic method G2^T of order w （w 〉 u） via any generalized linear multi-step method G3^T of the form ∑k=0^mαkZk = T∑k=0^mβkJ^-1↓ΔH（∑l=0^mγklZl）. We also give a necessary condition for this kind of generalized linear multi-step methods to be conjugate-symplectic. We also demonstrate that these results can be easily extended to the case when G3^T is a more general operator.

关键词： Linear multi-step method Generalized linear multi-step method Step-transition operator Infinitesimally symplectic Conjugate-symplectic.

Convergence analysis of the formal energies of symplectic methods for Hamiltonian systems

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Science China Mathematics 2016年第2期59卷 379-396页

作者： ZHANG RuiLi tang yifa ZHU BeiBei TU XiongBiao ZHAO Yue lsec icmsecacademy of mathematics and systems sciencechinese academy of sciencesBeijing 100190China

Based on Feng's theory of formal vector fields and formal flows, we study the convergence problem of the formal energies of symplectic methods for Hamiltonian systems and give the clear growth of the coefficients in the formal energies. With the help of B-series and Bernoulli functions, we prove that in the formal energy of the mid-point rule, the coefficient sequence of the merging products of an arbitrarily given rooted tree and the bushy trees of height 1(whose subtrees are vertices), approaches 0 as the number of branches goes to ∞; in the opposite direction, the coefficient sequence of the bushy trees of height m(m ≥ 2), whose subtrees are all tall trees, approaches ∞ at large speed as the number of branches goes to +∞. The conclusion extends successfully to the modified differential equations of other Runge-Kutta methods. This disproves a conjecture given by tang et al.(2002), and implies:(1) in the inequality of estimate given by Benettin and Giorgilli(1994) for the terms of the modified formal vector fields, the high order of the upper bound is reached in numerous cases;(2) the formal energies/formal vector fields are nonconvergent in general case.

关键词： convergence analysis formal energy symplectic method Hamiltonian system bushy tree

Symmetric and symplectic methods for gyrocenter dynamics in time-independent magnetic fields

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International Journal of Modeling, Simulation, and Scientific Computing 2016年第2期7卷 139-151页

作者： Beibei Zhu Zhenxuan Hu yifa tang Ruili Zhang LSEC ICMSEC Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190P.R.China School of Software Fudan University Shanghai 201203P.R.China Department of Modern Physics and School of Nuclear Science and Technology University of Science and Technology of China Hefei 230026 P.R.China

We apply a second-order symmetric Runge–Kutta method and a second-order symplectic Runge–Kutta method directly to the gyrocenter dynamics which can be expressed as a noncanonical Hamiltonian *** numerical simulation results show the overwhelming superiorities of the two methods over a higher order nonsymmetric nonsymplectic Runge–Kutta method in long-term numerical accuracy and near energy ***,they are much faster than the midpoint rule applied to the canonicalized system to reach given precision.

关键词： Symmetric Runge-Kutta method symplectic Runge-Kutta method numerical accuracy near energy conservation

Poisson Integrators Based on Splitting Method for Poisson Systems

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Communications in Computational Physics 2022年第9期32卷 1129-1155页

作者： Beibei Zhu Lun Ji Aiqing Zhu yifa tang Department of Applied Mathematics School of Mathematics and PhysicsUniversity of Science and Technology BeijingBeijing 100083China LSEC ICMSECAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China

We propose Poisson integrators for the numerical integration of separable Poisson *** analyze three situations in which Poisson systems are separated in threeways and Poisson integrators can be constructed by using the *** results show that the Poisson integrators outperform the higher order non-Poisson integrators in terms of long-termenergy conservation and computational *** Poisson integrators are also shown to be more efficient than the canonicalized sympletic methods of the same order.

关键词： Poisson systems Poisson integrators splitting method energy conservation