We propose an artificial boundary method for solving the deterministic Kardar-Parisi-Zhang equation in one-,two-and three dimensional unbounded domains.The exact artificial boundary conditions are obtained on the arti...We propose an artificial boundary method for solving the deterministic Kardar-Parisi-Zhang equation in one-,two-and three dimensional unbounded domains.The exact artificial boundary conditions are obtained on the artificial boundaries.Then the original problems are reduced to equivalent problems in bounded domains.A fi-nite difference method is applied to solve the reduced problems,and some numerical examples are provided to show the effectiveness of the method.展开更多
In this paper the numerical solution of the two-dimensional sine-Gordon equation is studied.Split local artificial boundary conditions are obtained by the operator splitting method.Then the original problem is reduced...In this paper the numerical solution of the two-dimensional sine-Gordon equation is studied.Split local artificial boundary conditions are obtained by the operator splitting method.Then the original problem is reduced to an initial boundary value problem on a bounded computational domain,which can be solved by the finite differencemethod.Several numerical examples are provided to demonstrate the effectiveness and accuracy of the proposed method,and some interesting propagation and collision behaviors of the solitary wave solutions are observed.展开更多
In this paper we use an analytical-numerical approach to find,in a systematic way,new 1-soliton solutions for a discrete sine-Gordon system in one spatial dimension.Since the spatial domain is unbounded,the numerical ...In this paper we use an analytical-numerical approach to find,in a systematic way,new 1-soliton solutions for a discrete sine-Gordon system in one spatial dimension.Since the spatial domain is unbounded,the numerical scheme employed to generate these soliton solutions is based on the artificial boundary method.A large selection of numerical examples provides much insight into the possible shapes of these new 1-solitons.展开更多
In scientific applications from plasma to chemical kinetics, a wide range of temporal scales can present in a system of differential equations. A major difficulty is encountered due to the stiffness of the system and ...In scientific applications from plasma to chemical kinetics, a wide range of temporal scales can present in a system of differential equations. A major difficulty is encountered due to the stiffness of the system and it is required to develop fast numerical schemes that are able to access previously unattainable parameter regimes. In this work, we consider an initial-final value problem for a multi-scale singularly perturbed system of linear ordi- nary differential equations with discontinuous coefficients. We construct a tailored finite point method, which yields approximate solutions that converge in the maximum norm, uniformly with respect to the singular perturbation parameters, to the exact solution. A parameter-uniform error estimate in the maximum norm is also proved. The results of numerical experiments, that support the theoretical results, are reported.展开更多
In this work we improve and extend a technique named recursive doubling procedure developed by Yuan and Lu[J.Lightwave Technology 25(2007),3649-3656]for solving periodic array problems.It turns out that when the perio...In this work we improve and extend a technique named recursive doubling procedure developed by Yuan and Lu[J.Lightwave Technology 25(2007),3649-3656]for solving periodic array problems.It 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 bands.This 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.展开更多
For the backward diffusion equation,a stable discrete energy regularization algorithm is proposed.Existence and uniqueness of the numerical solution are given.Moreover,the error between the solution of the given backw...For the backward diffusion equation,a stable discrete energy regularization algorithm is proposed.Existence and uniqueness of the numerical solution are given.Moreover,the error between the solution of the given backward diffusion equation and the numerical solution via the regularization method can be estimated.Some numerical experiments illustrate the efficiency of the method,and its application in image deblurring.展开更多
We propose two variants of tailored finite point(TFP)methods for discretizing two dimensional singular perturbed eigenvalue(SPE)problems.A continuation method and an iterative method are exploited for solving discreti...We propose two variants of tailored finite point(TFP)methods for discretizing two dimensional singular perturbed eigenvalue(SPE)problems.A continuation method and an iterative method are exploited for solving discretized systems of equations to obtain the eigen-pairs of the SPE.We study the analytical solutions of two special cases of the SPE,and provide an asymptotic analysis for the solutions.The theoretical results are verified in the numerical experiments.The numerical results demonstrate that the proposed schemes effectively resolve the delta function like of the eigenfunctions on relatively coarse grid.展开更多
This paper presents two uniformly convergent numerical schemes for the two dimensional steady state discrete ordinates transport equation in the diffusive regime,which is valid up to the boundary and interface layers....This paper presents two uniformly convergent numerical schemes for the two dimensional steady state discrete ordinates transport equation in the diffusive regime,which is valid up to the boundary and interface layers.A five-point nodecentered and a four-point cell-centered tailored finite point schemes(TFPS)are introduced.The schemes first approximate the scattering coefficients and sources by piecewise constant functions and then use special solutions to the constant coefficient equation as local basis functions to formulate a discrete linear system.Numerically,both methods can not only capture the diffusion limit,but also exhibit uniform convergence in the diffusive regime,even with boundary layers.Numerical results show that the five-point scheme has first-order accuracy and the four-point scheme has second-order accuracy,uniformly with respect to the mean free path.Therefore a relatively coarse grid can be used to capture the two dimensional boundary and interface layers.展开更多
Detecting corrosion by electrical field can be modeled by a Cauchy problem of Laplace equation in annulus domain under the assumption that the thickness of the pipe is relatively small compared with the radius of the ...Detecting corrosion by electrical field can be modeled by a Cauchy problem of Laplace equation in annulus domain under the assumption that the thickness of the pipe is relatively small compared with the radius of the pipe.The interior surface of the pipe is inaccessible and the nondestructive detection is solely based on measurements from the outer layer.The Cauchy problem for an elliptic equation is a typical ill-posed problem whose solution does not depend continuously on the boundary data.In this work,we assume that the measurements are available on the whole outer boundary on an annulus domain.By imposing reasonable assumptions,the theoretical goal here is to derive the stabilities of the Cauchy solutions and an energy regularization method.Relationship between the proposed energy regularization method and the Tikhonov regularization with Morozov principle is also given.A novel numerical algorithm is proposed and numerical examples are given.展开更多
基金National Natural Science Foundation of China,Hong Kong Research Grants Council and FRG of Hong Kong Baptist University.
文摘We propose an artificial boundary method for solving the deterministic Kardar-Parisi-Zhang equation in one-,two-and three dimensional unbounded domains.The exact artificial boundary conditions are obtained on the artificial boundaries.Then the original problems are reduced to equivalent problems in bounded domains.A fi-nite difference method is applied to solve the reduced problems,and some numerical examples are provided to show the effectiveness of the method.
基金support form the National Natural Science Foundation of China(Grant No.10971116).
文摘In this paper the numerical solution of the two-dimensional sine-Gordon equation is studied.Split local artificial boundary conditions are obtained by the operator splitting method.Then the original problem is reduced to an initial boundary value problem on a bounded computational domain,which can be solved by the finite differencemethod.Several numerical examples are provided to demonstrate the effectiveness and accuracy of the proposed method,and some interesting propagation and collision behaviors of the solitary wave solutions are observed.
基金The research was supported in part by the Natural Sciences and Engineering Research Council(NSERC)of Canada,by Hong Kong Research Grants Council and FRG of Hong Kong Baptist University.
文摘In this paper we use an analytical-numerical approach to find,in a systematic way,new 1-soliton solutions for a discrete sine-Gordon system in one spatial dimension.Since the spatial domain is unbounded,the numerical scheme employed to generate these soliton solutions is based on the artificial boundary method.A large selection of numerical examples provides much insight into the possible shapes of these new 1-solitons.
基金Acknowledgments. H. Han was supported by the NSFC Project No. 10971116. M. Tang is supported by Natural Science Foundation of Shanghai under Grant No. 12ZR1445400.
文摘In scientific applications from plasma to chemical kinetics, a wide range of temporal scales can present in a system of differential equations. A major difficulty is encountered due to the stiffness of the system and it is required to develop fast numerical schemes that are able to access previously unattainable parameter regimes. In this work, we consider an initial-final value problem for a multi-scale singularly perturbed system of linear ordi- nary differential equations with discontinuous coefficients. We construct a tailored finite point method, which yields approximate solutions that converge in the maximum norm, uniformly with respect to the singular perturbation parameters, to the exact solution. A parameter-uniform error estimate in the maximum norm is also proved. The results of numerical experiments, that support the theoretical results, are reported.
文摘In this work we improve and extend a technique named recursive doubling procedure developed by Yuan and Lu[J.Lightwave Technology 25(2007),3649-3656]for solving periodic array problems.It 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 bands.This 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.
基金National Natural Science Foundation of China(No.10471073)。
文摘For the backward diffusion equation,a stable discrete energy regularization algorithm is proposed.Existence and uniqueness of the numerical solution are given.Moreover,the error between the solution of the given backward diffusion equation and the numerical solution via the regularization method can be estimated.Some numerical experiments illustrate the efficiency of the method,and its application in image deblurring.
基金the National Natural Science Foundation of China through NSFC No.11371218 and No.91330203the second author was supported by the National Science Council of Taiwan through NSC 102-2115-M005-005.
文摘We propose two variants of tailored finite point(TFP)methods for discretizing two dimensional singular perturbed eigenvalue(SPE)problems.A continuation method and an iterative method are exploited for solving discretized systems of equations to obtain the eigen-pairs of the SPE.We study the analytical solutions of two special cases of the SPE,and provide an asymptotic analysis for the solutions.The theoretical results are verified in the numerical experiments.The numerical results demonstrate that the proposed schemes effectively resolve the delta function like of the eigenfunctions on relatively coarse grid.
基金supported by the NSFC Project No.10971116.M.Tang is supported by Natural Science Foundation of Shanghai under Grant No.12ZR1445400Shanghai Pujiang Program 13PJ1404700+1 种基金supported in part by the National Natural Science Foundation of China under Grant DMS-11101278the Young Thousand Talents Program of China.
文摘This paper presents two uniformly convergent numerical schemes for the two dimensional steady state discrete ordinates transport equation in the diffusive regime,which is valid up to the boundary and interface layers.A five-point nodecentered and a four-point cell-centered tailored finite point schemes(TFPS)are introduced.The schemes first approximate the scattering coefficients and sources by piecewise constant functions and then use special solutions to the constant coefficient equation as local basis functions to formulate a discrete linear system.Numerically,both methods can not only capture the diffusion limit,but also exhibit uniform convergence in the diffusive regime,even with boundary layers.Numerical results show that the five-point scheme has first-order accuracy and the four-point scheme has second-order accuracy,uniformly with respect to the mean free path.Therefore a relatively coarse grid can be used to capture the two dimensional boundary and interface layers.
基金supported by a CERG Grant of Hong Kong Research Grant Council,a FRG grant of Hong Kong Baptist University,and was partially supported by the NSFC Project No.19971116.
文摘Detecting corrosion by electrical field can be modeled by a Cauchy problem of Laplace equation in annulus domain under the assumption that the thickness of the pipe is relatively small compared with the radius of the pipe.The interior surface of the pipe is inaccessible and the nondestructive detection is solely based on measurements from the outer layer.The Cauchy problem for an elliptic equation is a typical ill-posed problem whose solution does not depend continuously on the boundary data.In this work,we assume that the measurements are available on the whole outer boundary on an annulus domain.By imposing reasonable assumptions,the theoretical goal here is to derive the stabilities of the Cauchy solutions and an energy regularization method.Relationship between the proposed energy regularization method and the Tikhonov regularization with Morozov principle is also given.A novel numerical algorithm is proposed and numerical examples are given.
