The Finite-Difference Time-Domain (FDTD) method is a well-known technique for the analysis of quantum devices. It solves a discretized Schrodinger equation in an iterative process. However, the method provides only a ...The Finite-Difference Time-Domain (FDTD) method is a well-known technique for the analysis of quantum devices. It solves a discretized Schrodinger equation in an iterative process. However, the method provides only a second-order accurate numerical solution and requires that the spatial grid size and time step should satisfy a very restricted condition in order to prevent the numerical solution from diverging. In this article, we present a generalized FDTD method with absorbing boundary condition for solving the one-dimensional (1D) time-dependent Schr?dinger equation and obtain a more relaxed condition for stability. The generalized FDTD scheme is tested by simulating a particle moving in free space and then hitting an energy potential. Numerical results coincide with those obtained based on the theoretical analysis.展开更多
With the linear interpolation method, an improved absorbing boundary condition（ABC）is introduced and derived, which is suitable for the alternating-direction-implicit finite- difference time-domain （ADI-FDTD） meth...With the linear interpolation method, an improved absorbing boundary condition（ABC）is introduced and derived, which is suitable for the alternating-direction-implicit finite- difference time-domain （ADI-FDTD） method. The reflection of the ABC caused by both the truncated error and the phase velocity error is analyzed. Based on the phase velocity estimation and the nonuniform cell, two methods are studied and then adopted to improve the performance of the ABC. A calculation case of a rectangular waveguide which is a typical dispersive transmission line is carried out using the ADI-FDTD method with the improved ABC for evaluation. According to the calculated case, the comparison is given between the reflection coefficients of the ABC with and without the velocity estimation and also the comparison between the reflection coefficients of the ABC with and without the nonuniform processing. The reflection variation of the ABC under different time steps is also analyzed and the acceptable worsening will not obscure the improvement on the absorption. Numerical results obviously show that efficient improvement on the absorbing performance of the ABC is achieved based on these methods for the ADI-FDTD.展开更多
We apply the newly proposed double absorbing boundary condition（DABC）（Hagstrom et al., 2014） to solve the boundary reflection problem in seismic finite-difference（FD） modeling. In the DABC scheme, the local high...We apply the newly proposed double absorbing boundary condition（DABC）（Hagstrom et al., 2014） to solve the boundary reflection problem in seismic finite-difference（FD） modeling. In the DABC scheme, the local high-order absorbing boundary condition is used on two parallel artificial boundaries, and thus double absorption is achieved. Using the general 2D acoustic wave propagation equations as an example, we use the DABC in seismic FD modeling, and discuss the derivation and implementation steps in detail. Compared with the perfectly matched layer（PML）, the complexity decreases, and the stability and fl exibility improve. A homogeneous model and the SEG salt model are selected for numerical experiments. The results show that absorption using the DABC is considerably improved relative to the Clayton–Engquist boundary condition and nearly the same as that in the PML.展开更多
A new absorbing boundary condition (ABC) for frequency dependent finite difference time domain algorithm for the arbitrary dispersive media is presented. The concepts of the digital systems are introduced to the (F...A new absorbing boundary condition (ABC) for frequency dependent finite difference time domain algorithm for the arbitrary dispersive media is presented. The concepts of the digital systems are introduced to the (FD) 2TD method. On the basis of digital filter designing and vector algebra, the absorbing boundary condition under arbitrary angle of incidence are derived. The transient electromagnetic problems in two dimensions and three dimensions are calculated and the validity of the ABC is verified.展开更多
Outgoing waves arising from high-velocity impacts between soil and structure can be reflected by the conventional truncated boundaries.Absorbing boundary conditions(ABCs),to attenuate the energy of the outward waves,a...Outgoing waves arising from high-velocity impacts between soil and structure can be reflected by the conventional truncated boundaries.Absorbing boundary conditions(ABCs),to attenuate the energy of the outward waves,are necessary to ensure the proper representation of the kinematic field and the accurate quantification of impact forces.In this paper,damping layer and dashpot ABCs are implemented in the material point method(MPM)with slight adjustments.Benchmark scenarios of different dynamic problems are modelled with the ABCs configured.Feasibility of the ABCs is assessed through the velocity fluctuations at specific observation points and the impact force fluctuations on the structures.The impact forces predicted by the MPM with ABCs are verified by comparison with those estimated using a computational fluid dynamics approach.展开更多
It is well-known that artificial boundary conditions are crucial for the efficient and accurate computations of wavefields on unbounded domains. In this paper, we investigate stability analysis for the wave equation c...It is well-known that artificial boundary conditions are crucial for the efficient and accurate computations of wavefields on unbounded domains. In this paper, we investigate stability analysis for the wave equation coupled with the first and the second order absorbing boundary conditions. The computational scheme is also developed. The approach allows the absorbing boundary conditions to be naturally imposed, which makes it easier for us to construct high order schemes for the absorbing boundary conditions. A thirdorder Lagrange finite element method with mass lumping is applied to obtain the spatial discretization of the wave equation. The resulting scheme is stable and is very efficient since no matrix inversion is needed at each time step. Moreover, we have shown both abstract and explicit conditional stability results for the fully-discrete schemes. The results are helpful for designing computational parameters in computations. Numerical computations are illustrated to show the efficiency and accuracy of our method. In particular, essentially no boundary reflection is seen at the artificial boundaries.展开更多
With the development of numerical methods the numerical computations require higher and higher accuracy. This paper is devoted to the high-order local absorbing boundary conditions （ABCs） for heat equation. We prove...With the development of numerical methods the numerical computations require higher and higher accuracy. This paper is devoted to the high-order local absorbing boundary conditions （ABCs） for heat equation. We proved that the coupled system yields a stable problem between the obtained high-order local ABCs and the partial differential equation in the computational domain. This method has been used widely in wave propagation models only recently. We extend the spirit of the methodology to parabolic ones, which will become a basis to design the local ABCs for a class of nonlinear PDEs. Some numerical tests show that the new treatment is very efficient and tractable.展开更多
This paper employs finite element method to solve shallow water equations with absorbing boundary conditions(the third kind,mixed boundary conditions).It is of practical importance in the cases that the land boundarie...This paper employs finite element method to solve shallow water equations with absorbing boundary conditions(the third kind,mixed boundary conditions).It is of practical importance in the cases that the land boundaries of the coastal area are made of porous medium allowing sea water flow in or out.The absorbing boundary conditions are treated as natural boundary conditions in wave equation finite element model.The numerical results for rectangu- lar and quarterly annular harbors indicate that the numerical solutions agree very well with ana- lytic solutions,which are also given in this paper.It is found that the land boundary absorbabili- ty may be significant to long wave oscillations,such as tidal waves in coastal harbors.展开更多
The performances of absorbing boundary conditions （ABCs） in four widely used finite difference time domain （FDTD） methods, i.e. explicit, implicit, explicit staggered-time, and Chebyshev methods, for solving the t...The performances of absorbing boundary conditions （ABCs） in four widely used finite difference time domain （FDTD） methods, i.e. explicit, implicit, explicit staggered-time, and Chebyshev methods, for solving the time-dependent Schrodinger equation are assessed and compared. The computation efficiency for each approach is also evaluated. A typical evolution problem of a single Gaussian wave packet is chosen to demonstrate the performances of the four methods combined with ABCs. It is found that ABCs perfectly eliminate reflection in implicit and explicit staggered-time methods. However, small reflection still exists in explicit and Chebyshev methods even though ABCs are applied.展开更多
The paper is concerned with the numerical solution of Schr¨odinger equations on an unbounded spatial domain.High-order absorbing boundary conditions for one-dimensional domain are derived,and the stability of the...The paper is concerned with the numerical solution of Schr¨odinger equations on an unbounded spatial domain.High-order absorbing boundary conditions for one-dimensional domain are derived,and the stability of the reduced initial boundary value problem in the computational interval is proved by energy estimate.Then a second order finite difference scheme is proposed,and the convergence of the scheme is established as well.Finally,numerical examples are reported to confirm our error estimates of the numerical methods.展开更多
Boundary procedure is an important phenomenon in numerical simulation. To reduce or eliminate the spurious reflections significantly which is occurred in boundary is a challenging and vital approach. The appropriate a...Boundary procedure is an important phenomenon in numerical simulation. To reduce or eliminate the spurious reflections significantly which is occurred in boundary is a challenging and vital approach. The appropriate artificial numerical boundaries can be applied to eliminate the effect of unnecessary spurious reflections in case of the numerical simulations of wave propagation phenomena problems. Typically, to reduce the artificial reflections, the absorbing boundary conditions are necessary. In this paper, we overview and investigate the appropriate typical absorbing boundary conditions and analyzed the boundary effect of two dimensional wave equation numerically. Reflections over the wide-ranging incident angles are complicated to eliminate, but the absorbing boundary conditions that we have applied are computationally cost efficient, easy to apply and able to reduce reflections significantly. For numerical solution, finite difference method is applied to develop numerical scheme using 2D wave equation. Using the developed numerical scheme, we obtain the numerical solution of the governing equation as an initial boundary value problem and realize the qualitative behavior of the solution in infinite space. The finite difference numerical scheme has been investigated by developing MATLAB programming language code. Numerical results have been discussed and analyzed with presenting different qualitative behavior of the numerical scheme. The accuracy and efficiency of the numerical scheme has been illustrated. The stability analysis was discussed and verified stability condition. Using the numerical scheme and absorbing boundary conditions, the boundary effects and absorption of spurious reflection of boundary have been demonstrated.展开更多
In Part I and Part II of this paper initial-boundary value problems of the acoustic wave equation with absorbing boundary conditions are considered. Their finite element-finite difference computational schemes are pr...In Part I and Part II of this paper initial-boundary value problems of the acoustic wave equation with absorbing boundary conditions are considered. Their finite element-finite difference computational schemes are proposed. The stability of the schemes is discussed and the corresponding stability conditions are given. Part I and Part II concern the first- and the second-order absorbing boundary conditions, respectively. Finally, numerical results are presented in Part II to show the correctness of theoretical analysis. (Author abstract) 7 Refs.展开更多
文摘The Finite-Difference Time-Domain (FDTD) method is a well-known technique for the analysis of quantum devices. It solves a discretized Schrodinger equation in an iterative process. However, the method provides only a second-order accurate numerical solution and requires that the spatial grid size and time step should satisfy a very restricted condition in order to prevent the numerical solution from diverging. In this article, we present a generalized FDTD method with absorbing boundary condition for solving the one-dimensional (1D) time-dependent Schr?dinger equation and obtain a more relaxed condition for stability. The generalized FDTD scheme is tested by simulating a particle moving in free space and then hitting an energy potential. Numerical results coincide with those obtained based on the theoretical analysis.
基金The National Natural Science Foundation of China(No.60702027)the Free Research Fund of the National Mobile Communications Research Laboratory of Southeast University (No.2008B07)the National Basic Research Program of China(973 Program)(No.2007CB310603)
文摘With the linear interpolation method, an improved absorbing boundary condition（ABC）is introduced and derived, which is suitable for the alternating-direction-implicit finite- difference time-domain （ADI-FDTD） method. The reflection of the ABC caused by both the truncated error and the phase velocity error is analyzed. Based on the phase velocity estimation and the nonuniform cell, two methods are studied and then adopted to improve the performance of the ABC. A calculation case of a rectangular waveguide which is a typical dispersive transmission line is carried out using the ADI-FDTD method with the improved ABC for evaluation. According to the calculated case, the comparison is given between the reflection coefficients of the ABC with and without the velocity estimation and also the comparison between the reflection coefficients of the ABC with and without the nonuniform processing. The reflection variation of the ABC under different time steps is also analyzed and the acceptable worsening will not obscure the improvement on the absorption. Numerical results obviously show that efficient improvement on the absorbing performance of the ABC is achieved based on these methods for the ADI-FDTD.
基金supported by the National Nature Science Foundation of China(Grant No.U1262208)the Important National Science & Technology Specific Projects(Grant No.2011ZX05019-008)
文摘We apply the newly proposed double absorbing boundary condition（DABC）（Hagstrom et al., 2014） to solve the boundary reflection problem in seismic finite-difference（FD） modeling. In the DABC scheme, the local high-order absorbing boundary condition is used on two parallel artificial boundaries, and thus double absorption is achieved. Using the general 2D acoustic wave propagation equations as an example, we use the DABC in seismic FD modeling, and discuss the derivation and implementation steps in detail. Compared with the perfectly matched layer（PML）, the complexity decreases, and the stability and fl exibility improve. A homogeneous model and the SEG salt model are selected for numerical experiments. The results show that absorption using the DABC is considerably improved relative to the Clayton–Engquist boundary condition and nearly the same as that in the PML.
文摘A new absorbing boundary condition (ABC) for frequency dependent finite difference time domain algorithm for the arbitrary dispersive media is presented. The concepts of the digital systems are introduced to the (FD) 2TD method. On the basis of digital filter designing and vector algebra, the absorbing boundary condition under arbitrary angle of incidence are derived. The transient electromagnetic problems in two dimensions and three dimensions are calculated and the validity of the ABC is verified.
基金the Key Science and Technology Plan of Power China Huadong Engineering Corporation(No.KY2018-ZD-01)China and the National Natural Science Foundations of China(No.51909248)。
文摘Outgoing waves arising from high-velocity impacts between soil and structure can be reflected by the conventional truncated boundaries.Absorbing boundary conditions(ABCs),to attenuate the energy of the outward waves,are necessary to ensure the proper representation of the kinematic field and the accurate quantification of impact forces.In this paper,damping layer and dashpot ABCs are implemented in the material point method(MPM)with slight adjustments.Benchmark scenarios of different dynamic problems are modelled with the ABCs configured.Feasibility of the ABCs is assessed through the velocity fluctuations at specific observation points and the impact force fluctuations on the structures.The impact forces predicted by the MPM with ABCs are verified by comparison with those estimated using a computational fluid dynamics approach.
文摘It is well-known that artificial boundary conditions are crucial for the efficient and accurate computations of wavefields on unbounded domains. In this paper, we investigate stability analysis for the wave equation coupled with the first and the second order absorbing boundary conditions. The computational scheme is also developed. The approach allows the absorbing boundary conditions to be naturally imposed, which makes it easier for us to construct high order schemes for the absorbing boundary conditions. A thirdorder Lagrange finite element method with mass lumping is applied to obtain the spatial discretization of the wave equation. The resulting scheme is stable and is very efficient since no matrix inversion is needed at each time step. Moreover, we have shown both abstract and explicit conditional stability results for the fully-discrete schemes. The results are helpful for designing computational parameters in computations. Numerical computations are illustrated to show the efficiency and accuracy of our method. In particular, essentially no boundary reflection is seen at the artificial boundaries.
文摘With the development of numerical methods the numerical computations require higher and higher accuracy. This paper is devoted to the high-order local absorbing boundary conditions （ABCs） for heat equation. We proved that the coupled system yields a stable problem between the obtained high-order local ABCs and the partial differential equation in the computational domain. This method has been used widely in wave propagation models only recently. We extend the spirit of the methodology to parabolic ones, which will become a basis to design the local ABCs for a class of nonlinear PDEs. Some numerical tests show that the new treatment is very efficient and tractable.
文摘This paper employs finite element method to solve shallow water equations with absorbing boundary conditions(the third kind,mixed boundary conditions).It is of practical importance in the cases that the land boundaries of the coastal area are made of porous medium allowing sea water flow in or out.The absorbing boundary conditions are treated as natural boundary conditions in wave equation finite element model.The numerical results for rectangu- lar and quarterly annular harbors indicate that the numerical solutions agree very well with ana- lytic solutions,which are also given in this paper.It is found that the land boundary absorbabili- ty may be significant to long wave oscillations,such as tidal waves in coastal harbors.
基金supported by the State Key Development Program for Basic Research of China (No. 2006CB932404)
文摘The performances of absorbing boundary conditions （ABCs） in four widely used finite difference time domain （FDTD） methods, i.e. explicit, implicit, explicit staggered-time, and Chebyshev methods, for solving the time-dependent Schrodinger equation are assessed and compared. The computation efficiency for each approach is also evaluated. A typical evolution problem of a single Gaussian wave packet is chosen to demonstrate the performances of the four methods combined with ABCs. It is found that ABCs perfectly eliminate reflection in implicit and explicit staggered-time methods. However, small reflection still exists in explicit and Chebyshev methods even though ABCs are applied.
基金supported by FRG of Hong Kong Baptist University,RGC of Hong Kong,Natural Science Foundation of China(Grant Number 10871044)Singapore AcRF RG59/08(M52110092)NRF 2007IDM-IDM002-010.
文摘The paper is concerned with the numerical solution of Schr¨odinger equations on an unbounded spatial domain.High-order absorbing boundary conditions for one-dimensional domain are derived,and the stability of the reduced initial boundary value problem in the computational interval is proved by energy estimate.Then a second order finite difference scheme is proposed,and the convergence of the scheme is established as well.Finally,numerical examples are reported to confirm our error estimates of the numerical methods.
文摘Boundary procedure is an important phenomenon in numerical simulation. To reduce or eliminate the spurious reflections significantly which is occurred in boundary is a challenging and vital approach. The appropriate artificial numerical boundaries can be applied to eliminate the effect of unnecessary spurious reflections in case of the numerical simulations of wave propagation phenomena problems. Typically, to reduce the artificial reflections, the absorbing boundary conditions are necessary. In this paper, we overview and investigate the appropriate typical absorbing boundary conditions and analyzed the boundary effect of two dimensional wave equation numerically. Reflections over the wide-ranging incident angles are complicated to eliminate, but the absorbing boundary conditions that we have applied are computationally cost efficient, easy to apply and able to reduce reflections significantly. For numerical solution, finite difference method is applied to develop numerical scheme using 2D wave equation. Using the developed numerical scheme, we obtain the numerical solution of the governing equation as an initial boundary value problem and realize the qualitative behavior of the solution in infinite space. The finite difference numerical scheme has been investigated by developing MATLAB programming language code. Numerical results have been discussed and analyzed with presenting different qualitative behavior of the numerical scheme. The accuracy and efficiency of the numerical scheme has been illustrated. The stability analysis was discussed and verified stability condition. Using the numerical scheme and absorbing boundary conditions, the boundary effects and absorption of spurious reflection of boundary have been demonstrated.
文摘In Part I and Part II of this paper initial-boundary value problems of the acoustic wave equation with absorbing boundary conditions are considered. Their finite element-finite difference computational schemes are proposed. The stability of the schemes is discussed and the corresponding stability conditions are given. Part I and Part II concern the first- and the second-order absorbing boundary conditions, respectively. Finally, numerical results are presented in Part II to show the correctness of theoretical analysis. (Author abstract) 7 Refs.
