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Numerical Simulation of Rogue Waves by the Local Discontinuous galerkin Method

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Chinese Physics Letters 2014年第4期31卷 1-4页

作者： CAI Wen-Jun WANG Yu-Shun SONG Yong-Zhong Jiangsu Provincial Key Laboratory for NSLSCS School of Mathematics and Sciences Nanjing Normal University Nanjing 210023

We study rogue waves described by nonlinear Schr6dinger equations. Such wave solutions are so different from conventional soliton solutions that classic methods such as the Crank-Nicolson scheme cannot work for these cases. Fortunately, we find that the local discontinuous galerkin method equipped with Dirichlet boundary conditions can simulate rogue waves very well. Several numerical examples are presented to show such interesting wave solutions.

关键词： ROGUE waves SCHRODINGER equation SOLITONS galerkin methods DIRICHLET forms

MsFEM à la Crouzeix-Raviart for Highly Oscillatory Elliptic Problems

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Chinese Annals of Mathematics,Series B 2013年第1期34卷 113-138页

作者： Claude LE BRIS Frédéric LEGOLL Alexei LOZINSKI cole Nationale des Ponts et Chaussées 6 et 8 avenue Blaise Pascal INRIA Rocquencourt MICMAC project-team Institut de Mathématiques de Toulouse Université Paul Sabatier Laboratoire de Mathématiques CNRS UMR 6623 Université de Franche-Comté

We introduce and analyze a multiscale finite element type method (MsFEM) in the vein of the classical Crouzeix-Raviart finite element method that is specifically adapted for highly oscillatory elliptic problems. We il... 详细信息

关键词： Homogenization Finite elements galerkin methods Highly oscillatoryPDE

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Intermittency and Thermalization in Turbulence

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Chinese Physics Letters 2010年第5期27卷 152-155页

作者：朱建州 MarkTaylor Theoretical Division and CNLS MS B258 Los Alamos National Laboratory Los Alamos NM 87545 U.S.A. CCIM MS 0370 Sandia National Laboratories Albuquerque NM 87185 U.S.A.

A dissipation rate, which grows faster than any power of the wave number in Fourier space, may be scaled to lead a hydrodynamic system to actually or potentially converge to its galerkin truncation. Actual convergence here means the asymptotic truncation at a finite wavenumber kc above which modes have no dynamics; and, we define potential convergence for the truncation at kG which, however, grows without bound. Both types of convergence can be obtained with the dissipation rate μ[cosh（k/kc） - 1] that behaves as k2 （newtonian） and exp{k/kc} for small and large k/kc respectively. Competing physics of cascade, thermalization and dissipation are discussed for numerical Navier-Stokes turbulence, emphasizing the intermittency growth issue.

关键词： THERMAL analysis TURBULENCE ENERGY dissipation FOURIER transforms HYDRODYNAMICS STOCHASTIC convergence galerkin methods ASYMPTOTIC expansions NAVIER-Stokes equations

NONLINEAR galerkin METHOD FOR THE EXTERIORNONSTATIONARY NAVIER-STOKES EQUATIONS

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Applied Mathematics and Mechanics(English Edition) 2002年第11期23卷 1282-1291页

作者： HE Yin-nian(何银年) LI Kai-tai(李开泰) Department of Scientific Computing Faculty of Science Xi'' an Jiaotong University Xi''an 710049 P R China Department of Scientific Computing Faculty of Science Xi'' an Jiaotong University Xi''an 710049 P R China

A new algorithm combining nonlinear galerkin method and coupling method of finite element and boundary element is introduced to solve the exterior nonstationary Navier-Stokes equations. The regularity of the coupling variational formulation and the convergence of the approximate solution corresponding to the algorithm are proved. If the fine mesh h is chosen as coarse mesh H-square, the nonlinear galerkin method, nonlinearity is only treated on the coarse grid and linearity is treated on the fine grid. Hence, the new algorithm can save a large amount of computational time.

关键词： Boundary element method Convergence of numerical methods Finite element method galerkin methods Nonlinear equations Variational techniques

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On Discontinuous and Continuous Approximations to Second-Kind Volterra Integral Equations

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Numerical Mathematics(Theory,methods and Applications) 2022年第1期15卷 91-124页

作者： Hui Liang School of Science Harbin Institute of TechnologyShenzhen518055China

Collocation and galerkin methods in the discontinuous and globally continuous piecewise polynomial spaces,in short,denoted as DC,CC,DG and CG methods respectively,are employed to solve second-kind Volterra integral equations(VIEs).It is proved that the quadrature DG and CG(QDG and QCG)methods obtained from the DG and CG methods by approximating the inner products by suitable numerical quadrature formulas,are equivalent to the DC and CC methods,*** addition,the fully discretised DG and CG(FDG and FCG)methods are equivalent to the corresponding fully discretised DC and CC(FDC and FCC)*** convergence theories are established for DG and CG methods,and their semi-discretised(QDG and QCG)and fully discretized(FDG and FCG)*** particular,it is proved that the CG method for second-kind VIEs possesses a similar convergence to the DG method for first-kind *** examples illustrate the theoretical results.

关键词： Volterra integral equations collocation methods galerkin methods discontinuous galerkin methods convergence analysis

Adaptive Finite Element Modeling Techniques for the Poisson-Boltzmann Equation

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Communications in Computational Physics 2012年第1期11卷 179-214页

作者： M.Holst J.A.McCammon Z.Yu Y.C.Zhou Y.Zhu Department of Mathematics University of California at San DiegoLa JollaCA 92093USA Department of Physics University of California at San DiegoLa JollaCA 92093USA Department of Chemistry&Biochemistry University of California at San DiegoLa JollaCA 92093USA Center for Theoretical Biological Physics(CTBP) University of California at San DiegoLa JollaCA 92093USA National Biomedical Computation Resource(NBCR) University of California at San DiegoLa JollaCA 92093USA Howard Hughes Medical Institute University of California at San DiegoLa JollaCA 92093USA

We consider the design of an effective and reliable adaptive finite element method(AFEM)for the nonlinear Poisson-Boltzmann equation(PBE).We first examine the two-term regularization technique for the continuous problem recently proposed by Chen,Holst and Xu based on the removal of the singular electrostatic potential inside biomolecules;this technique made possible the development of the first complete solution and approximation theory for the Poisson-Boltzmann equation,the first provably convergent discretization and also allowed for the development of a provably convergent ***,in practical implementation,this two-term regularization exhibits numerical ***,we examine a variation of this regularization technique which can be shown to be less susceptible to such *** establish a priori estimates and other basic results for the continuous regularized problem,as well as for galerkin finite element *** show that the new approach produces regularized continuous and discrete problemswith the samemathematical advantages of the original *** then design an AFEM scheme for the new regularized problem and show that the resulting AFEM scheme is accurate and reliable,by proving a contraction result for the *** result,which is one of the first results of this type for nonlinear elliptic problems,is based on using continuous and discrete a priori L¥*** provide a high-quality geometric model as input to the AFEM algorithm,we also describe a class of feature-preserving adaptive mesh generation algorithms designed specifically for constructing meshes of biomolecular structures,based on the intrinsic local structure tensor of the molecular *** of the algorithms described in the article are implemented in the Finite Element Toolkit(FETK),developed and maintained at *** stability advantages of the new regularization scheme are demonstrated with FETK through comparisons with the original regula

关键词： Poisson-Boltzmann equation semi-linear partial differential equations supercritical nonlinearity singularity a priori L¥estimates existence uniqueness well-posedness galerkin methods discrete a priori L¥estimates quasi-optimal a priori error estimates adaptive finite methods contraction convergence optimality surface and volume mesh generation mesh improvement and decimation.

Handbook of sinc numerical methods

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Energy-Preserving H1–galerkin Schemes for the Hunter-Saxton Equation

丛书名： Chapman & Hall/CRC numerical analysis and scientific computing

2011年

作者： Stenger Frank.

ISBN: (纸本)9781439821589;1439821585

"This handbook is essential for solving numerical problems in mathematics, computer science, and engineering. The methods presented are similar to finite elements but more adept at solving analytic problems with singularities over irregularly shaped yet analytically described regions. The author makes sinc methods accessible to potential users by limiting details as to how or why these methods wor

关键词： Differential equations galerkin methods

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Journal of Mathematical Research with Applications 2017年第1期37卷 107-118页

作者： Yuto MIYATAKE Geonsik EOM Tomohiro SOGABE Shaoliang ZHANG Department of Computational Science and Engineering Graduate School of EngineeringNagoya University

We consider the numerical integration of the Hunter–Saxton equation, which models the propagation of weakly nonlinear orientation waves. For the equation, we present two weak forms and their galerkin discretizations. The galerkin schemes preserve the Hamiltonian of the equation and can be implemented with cheap H1elements. Numerical experiments confirm the effectiveness of the schemes.

关键词： Hunter-Saxton equation energy-preservation galerkin methods