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APPROXIMATION, STABILITY AND FAST EVALUATION OF EXACT ARTIFICIAL BOUNDARY CONDITION FOR THE ONE-DIMENSIONAL HEAT EQUATION

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Journal of Computational Mathematics 2007年第6期25卷 730-745页

作者： chunxiong zheng Department of Mathematical Sciences Tsinghua University Beijing 100084 China Institut für Numerische Mathematik Universitaet Miinster Germany

In this paper we consider the numerical solution of the one-dimensional heat equation on unbounded domains. First an exact semi-discrete artificial boundary condition is derived by discretizing the time variable with the Crank-Nicolson method. The semi-discretized heat equation equipped with this boundary condition is then proved to be unconditionally stable, and its solution is shown to have second-order accuracy. In order to reduce the computational cost, we develop a new fast evaluation method for the convolution operation involved in the exact semi-discrete artificial boundary condition. A great advantage of this method is that the unconditional stability held by the semi-discretized heat equation is preserved. An error estimate is also given to show the dependence of numerical errors on the time step and the approximation accuracy of the convolution kernel. Finally, a simple numerical example is presented to validate the theoretical results.

关键词： Heat equation Artificial boundary conditions Fast evaluation Unbounded domains.

An Exact Absorbing Boundary Condition for the Schr¨odinger Equation With Sinusoidal Potentials at Infinity

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Communications in Computational Physics 2008年第3期3卷 641-658页

作者： chunxiong zheng Department of Mathematical Sciences Tsinghua UniversityBeijing 100084China.

In this paper we study numerical issues related to the Schr ¨odinger equationwith sinusoidal potentials at infinity. An exact absorbing boundary condition in a formof Dirichlet-to-Neumann mapping is derived. This boundary condition is based on ananalytical expression of the logarithmic derivative of the Floquet solution toMathieu’sequation, which is completely new to the author’s knowledge. The implementationof this exact boundary condition is discussed, and a fast evaluation method is used toreduce the computation burden arising from the involved half-order derivative *** numerical tests are given to showthe performance of the proposed absorbingboundary conditions.

关键词： Absorbing boundary condition sinusoidal potential Schr¨odinger equation unbounded domain.

POISSON PRECONDITIONING FOR SELF-ADJOINT ELLIPTIC PROBLEMS

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Journal of Computational Mathematics 2014年第5期32卷 560-578页

作者： Houde Han chunxiong zheng Department of Mathematical Sciences Tsinghua University Beijing 100084 China

In this paper, we formulate interface problem and Neumann elliptic boundary value problem into a form of linear operator equations with self-adjoint positive definite op- erators. We prove that in the discrete level the condition number of these operators is independent of the mesh size. Therefore, given a prescribed error tolerance, the classical conjugate gradient algorithm converges within a fixed number of iterations. The main computation task at each iteration is to solve a Dirichlet Poisson boundary value problem in a rectangular domain, which can be furnished with fast Poisson solver. The overall computational complexity is essentially of linear scaling.

关键词： Fast Poisson solver Interface problem Self-adjoint elliptic problem Conjugategradient method.

A Time-Splitting Spectral Method for Three-Wave Interactions in Media with Competing Quadratic and Cubic Nonlinearities

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Communications in Computational Physics 2007年第1期2卷 123-140页

作者： Weizhu Bao chunxiong zheng Department of Mathematics National University of SingaporeSingapore 117543 Department of Mathematical Sciences Tsinghua UniversityBeijing100084P.R.China.

This paper introduces an extension of the time-splitting spectral(TSSP)method for solving a general model of three-wave optical interactions,which typically arises from nonlinear optics,when the transmission media has competing quadratic and cubic *** key idea is to formulate the terms related to quadratic and cubic nonlinearities into a Hermitian matrix in a proper way,which allows us to develop an explicit and unconditionally stable numerical method for the ***,the method is spectral accurate in transverse coordinates and second-order accurate in propagation direction,is time reversible and time transverse invariant,and conserves the total wave energy(or power or the norm of the solutions)in discretized *** examples are presented to demonstrate the efficiency and high resolution of the *** the method is applied to study dynamics and interactions between three-wave solitons and continuous waves in media with competing quadratic and cubic nonlinearities in one dimension(1D)and 2D.

关键词： Nonlinear optics three-wave time-splitting spectral method energy continuous wave soliton.

Composite Coherent States Approximation for One-Dimensional Multi-Phased Wave Functions

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Communications in Computational Physics 2012年第3期11卷 951-984页

作者： Dongsheng Yin chunxiong zheng Department of Mathematical Sciences Tsinghua UniversityBeijing100084P.R.China

The coherent states approximation for one-dimensional multi-phased wave functions is considered in this *** wave functions are assumed to oscillate on a characteristic wave length O(∈)withǫ≪1.A parameter recovery algorithm is first developed for single-phased wave function based on a moment asymptotic *** algorithm is then extended to multi-phased wave *** cross points or caustics exist,the coherent states approximation algorithm based on the parameter recovery will fail in some local *** this case,we resort to the windowed Fourier transform technique,and propose a composite coherent states approximation *** experiments show that the number of coherent states derived by the proposedmethod is much less than that by the directwindowed Fourier transform technique.

关键词： Coherent state high frequency windowed Fourier transform

Global geometrical optics method for vector-valued Schrodinger problems

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Frontiers of Mathematics in China 2018年第3期13卷 579-606页

作者： Jiashun HU Xiang MA chunxiong zheng Department of Mathematical Sciences Tsinghua University Beijing 100084 China

We extend the theory of global geometrical optics method, proposed originally for the linear scalar high-frequency wave-like equations in [Commun. Math. Sci., 2013, 11（1）： 105-140], to the more general vector- valued Schrodinger problems in the semi-classical regime. The key ingredient in the global geometrical optics method is a moving frame technique in the phase space. The governing equation is transformed into a new equation but of the same type when expressed in any moving frame induced by the underlying Hamiltonian flow. The classical Wentzel-Kramers-Brillouin （WKB） analysis benefits from this treatment as it maintains valid for arbitrary but fixed evolutionary time. It turns out that a WKB-type function defined merely on the underlying Lagrangian submanifold can be obtained with the help of this moving frame technique, and from which a uniform first-order approximation of the wave field can be derived, even around caustics. The general theory is exemplified by two specific instances. One is the two-level SchrSdinger system and the other is the periodic SchrSdinger equation. Numerical tests validate the theoretical results.

关键词： Global geometrical optics method Hamiltonian system unitary representation caustics semiclassical approximation

Dirichlet-to-Neumann Mapping for the Characteristic Elliptic Equations with Symmetric Periodic Coefficients

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Communications in Computational Physics 2014年第9期16卷 1102-1115页

作者： Jingsu Kang Meirong Zhang chunxiong zheng Department of Mathematical Sciences Tsinghua UniversityBeijing 100084P.R.China.

Based on the numerical evidences,an analytical expression of the Dirichletto-Neumann mapping in the form of infinite product was first conjectured for the onedimensional characteristic Schrodinger equation with a sinusoidal potential in[***.,3(3):641-658,2008].It was later extended for the general secondorder characteristic elliptic equations with symmetric periodic coefficients in[***.,227:6877-6894,2008].In this paper,we present a proof for this Dirichlet-toNeumann mapping.

关键词： Dirichlet-to-Neumann mapping Schrodinger equation symmetric periodic potentials absorbing boundary conditions.

Numerical Computations of Nonlocal Schrodinger Equations on the Real Line

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Communications on Applied Mathematics and Computation 2020年第2期2卷 241-260页

作者： Yonggui Yan Jiwei Zhang chunxiong zheng Beijing Computational Science Research Center Beijing 100093China School of Mathematics and Statistics and Hubei Key Laboratory of Computational ScienceWuhan UniversityWuhan 430072China Department of Mathematical Sciences Tsinghua UniversityBeijing 100084China College of Mathematics and Systems Science Xinjiang UniversityUnimqi 830046China

The numerical computation of nonlocal Schrödinger equations (SEs) on the whole real axis is considered. Based on the artifcial boundary method, we frst derive the exact artifcial nonrefecting boundary conditions. For the numerical implementation, we employ the quadrature scheme proposed in Tian and Du (SIAM J Numer Anal 51:3458-3482, 2013) to discretize the nonlocal operator, and apply the z-transform to the discrete nonlocal system in an exterior domain, and derive an exact solution expression for the discrete system. This solution expression is referred to our exact nonrefecting boundary condition and leads us to reformulate the original infnite discrete system into an equivalent fnite discrete system. Meanwhile, the trapezoidal quadrature rule is introduced to discretize the contour integral involved in exact boundary conditions. Numerical examples are fnally provided to demonstrate the efectiveness of our approach.

关键词： Nonrefecting boundary conditions Artifcial boundary method Nonlocal Schrödinger equation z-transform Nonlocal models

Numerical Simulation of Waves in Periodic Structures

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Communications in Computational Physics 2009年第5期5卷 849-870页

作者： Matthias Ehrhardt Houde Han chunxiong zheng Weierstraß-Institut fur Angewandte Analysis und Stochastik Mohrenstr.3910117 BerlinGermany Department of Mathematical Sciences Tsinghua UniversityBeijing 100084China

In this work we improve and extend a technique named recursive doubling procedure developed by Yuan and Lu[*** Technology 25(2007),3649-3656]for solving periodic array *** turns out that when the periodic array contains an infinite number of periodic cells,our method gives a fast evaluation of the exact boundary Robin-to-Robin mapping if the wave number is complex,or real but in the stop *** technique is also used to solve the time-dependent Schr¨odinger equation in both one and two dimensions,when the periodic potential functions have some local defects.

关键词： Periodic media Helmholtz equation Schrodinger equation Dirichlet-to-Neumann maps Robin-to-Robin maps band structure Floquet-Bloch theory high-order finite elements