A modified alternating direction implicit algorithm is proposed to solve the full-vectorial finite-difference beam propagation method formulation based on H fields. The cross-coupling terms are neglected in the first ...A modified alternating direction implicit algorithm is proposed to solve the full-vectorial finite-difference beam propagation method formulation based on H fields. The cross-coupling terms are neglected in the first sub-step, but evaluated and doubly used in the second sub-step. The order of two sub-steps is reversed for each transverse magnetic field component so that the cross-coupling terms are always expressed in implicit form, thus the calculation is very efficient and stable. Moreover, an improved six-point finite-difference scheme with high accuracy independent of specific structures of waveguide is also constructed to approximate the cross-coupling terms along the transverse directions. The imaginary-distance procedure is used to assess the validity and utility of the present method. The field patterns and the normalized propagation constants of the fundamental mode for a buried rectangular waveguide and a rib waveguide are presented. Solutions are in excellent agreement with the benchmark results from the modal transverse resonance method.展开更多
Alternating direction implicit finite difference time domain (ADI-FDTD) method is unconditionally stable and the maximum time step is not limited by the Courant stability condition, but rather by numerical error. Co...Alternating direction implicit finite difference time domain (ADI-FDTD) method is unconditionally stable and the maximum time step is not limited by the Courant stability condition, but rather by numerical error. Compared with the conventional FDTD method, the time step of ADI-FDTD can be enlarged arbitrarily and the CPU cost can be reduced. 2D perfectly matched layer (PML) absorbing boundary condition is proposed to truncate computation space for ADI-FDTD in dispersive media using recursive convolution(RC) method and the 2D PML formulations for dispersive media are derived. ADI-FDTD formulations for dispersive media can be obtained from the simplified PML formulations. The scattering of target in dispersive soil is simulated under sine wave and Gaussian pulse excitations and numerical results of ADI-FDTD with PML are compared with FDTD. Good agreement is observed. At the same time the CPU cost for ADI-FDTD is obviously reduced.展开更多
In this paper,two classes of Riesz space fractional partial differential equations including space-fractional and space-time-fractional ones are considered.These two models can be regarded as the generalization of the...In this paper,two classes of Riesz space fractional partial differential equations including space-fractional and space-time-fractional ones are considered.These two models can be regarded as the generalization of the classical wave equation in two space dimensions.Combining with the Crank-Nicolson method in temporal direction,efficient alternating direction implicit Galerkin finite element methods for solving these two fractional models are developed,respectively.The corresponding stability and convergence analysis of the numerical methods are discussed.Numerical results are provided to verify the theoretical analysis.展开更多
Numerical simulation of complex flow fields with multi-scale structures is one of the most important and challenging branches of computational fluid dynamics. From linear analysis and numerical experiments it has been...Numerical simulation of complex flow fields with multi-scale structures is one of the most important and challenging branches of computational fluid dynamics. From linear analysis and numerical experiments it has been discovered that the higher-order accurate method can give reliable and efficient computational results, as well as better resolution of the complex flow fields with multi-scale structures. Compact finite difference schemes, which feature higher-order accuracy and spectral-like resolution with smaller stencils and easier application of boundary conditions, has attracted more and more interest and attention.展开更多
A parabolic equation (PE) based method for analyzing composite scattering under an electromagnetic wave incidence at low grazing angle, which composes of three-dimensional (3-D) electrically large targets and roug...A parabolic equation (PE) based method for analyzing composite scattering under an electromagnetic wave incidence at low grazing angle, which composes of three-dimensional (3-D) electrically large targets and rough surface, is presented and discussed. A superior high-order PE version is used to improve the accuracy at wider paraxial angles, and along with the alternating direction implicit (ADI) differential technique, the computational efficiency is further improved. The formula of bistatic normalized radar cross section is derived by definition and near-far field transformation. Numerical examples are given to show the validity and accuracy of the proposed approach, in which the results are compared with those of Kirchhoff approximation (KA) and moment of method (MoM). Furthermore, the bistatic scattering properties of composite model in which the 3-D PEC targets on or above the two-dimensional Gaussian rough surfaces under the tapered wave incidence are analyzed.展开更多
An efficient high-order immersed interface method (IIM) is proposed to solve two-dimensional (2D) heat problems with fixed interfaces on Cartesian grids, which has the fourth-order accuracy in the maximum norm in ...An efficient high-order immersed interface method (IIM) is proposed to solve two-dimensional (2D) heat problems with fixed interfaces on Cartesian grids, which has the fourth-order accuracy in the maximum norm in both time and space directions. The space variable is discretized by a high-order compact (HOC) difference scheme with correction terms added at the irregular points. The time derivative is integrated by a Crank-Nicolson and alternative direction implicit (ADI) scheme. In this case, the time accuracy is just second-order. The Richardson extrapolation method is used to improve the time accuracy to fourth-order. The numerical results confirm the convergence order and the efficiency of the method.展开更多
New numerical techniques are presented for the solution of the twodimensional time fractional evolution equation in the unit square.In these methods,Galerkin finite element is used for the spatial discretization,and,f...New numerical techniques are presented for the solution of the twodimensional time fractional evolution equation in the unit square.In these methods,Galerkin finite element is used for the spatial discretization,and,for the time stepping,new alternating direction implicit(ADI)method based on the backward Euler method combined with the first order convolution quadrature approximating the integral term are considered.The ADI Galerkin finite element method is proved to be convergent in time and in the L2 norm in space.The convergence order is O(k|ln k|+h^(r)),where k is the temporal grid size and h is spatial grid size in the x and y directions,respectively.Numerical results are presented to support our theoretical analysis.展开更多
Based on rectangular partition and bilinear interpolation,we construct an alternating-direction implicit(ADI)finite volume element method,which combined the merits of finite volume element method and alternating direc...Based on rectangular partition and bilinear interpolation,we construct an alternating-direction implicit(ADI)finite volume element method,which combined the merits of finite volume element method and alternating direction implicit method to solve a viscous wave equation with variable coefficients.This paper presents a general procedure to construct the alternating-direction implicit finite volume element method and gives computational schemes.Optimal error estimate in L2 norm is obtained for the schemes.Compared with the finite volume element method of the same convergence order,our method is more effective in terms of running time with the increasing of the computing scale.Numerical experiments are presented to show the efficiency of our method and numerical results are provided to support our theoretical analysis.展开更多
An alternating direction implicit (ADI) Galerkin method with moving finite element spaces is formulated for a class of second order hyperbolic equations in two space variables. A priori H 1 error estimate is derived.
In this paper, a two-dimensional nonlinear coupled Gray Scott system is simulated with a finite difference scheme and a finite volume technique. Pre and post-processing lead to a new solution called GSmFoam by underst...In this paper, a two-dimensional nonlinear coupled Gray Scott system is simulated with a finite difference scheme and a finite volume technique. Pre and post-processing lead to a new solution called GSmFoam by understandin<span>g geometry settings and mesh information. The concentration profile chan</span>ges over time, as does the intensity of the contour patterns. The OpenFoam solver gives you the confidence to compare the pattern result with efficient numerical algorithms on the Gray Scott model.展开更多
In this paper, an alternating direction Galerkin finite element method is presented for solving 2D time fractional reaction sub-diffusion equation with nonlinear source term. Firstly, one order implicit-explicit metho...In this paper, an alternating direction Galerkin finite element method is presented for solving 2D time fractional reaction sub-diffusion equation with nonlinear source term. Firstly, one order implicit-explicit method is used for time discretization, then Galerkin finite element method is adopted for spatial discretization and obtain a fully discrete linear system. Secondly, Galerkin alternating direction procedure for the system is derived by adding an extra term. Finally, the stability and convergence of the method are analyzed rigorously. Numerical results confirm the accuracy and efficiency of the proposed method.展开更多
This paper detailedly discusses the locally one-dimensional numerical methods for ef- ficiently solving the three-dimensional fractional partial differential equations, including fractional advection diffusion equatio...This paper detailedly discusses the locally one-dimensional numerical methods for ef- ficiently solving the three-dimensional fractional partial differential equations, including fractional advection diffusion equation and Riesz fractional diffusion equation. The second order finite difference scheme is used to discretize the space fractional derivative and the Crank-Nicolson procedure to the time derivative. We theoretically prove and numerically verify that the presented numerical methods are unconditionally stable and second order convergent in both space and time directions. In particular, for the Riesz fractional dif- fusion equation, the idea of reducing the splitting error is used to further improve the algorithm, and the unconditional stability and convergency are also strictly proved and numerically verified for the improved scheme.展开更多
In this paper,we consider the numerical solutions of the semilinear Riesz space-fractional diffusion equations(RSFDEs)with time delay,which constitute an important class of differential equations of practical signific...In this paper,we consider the numerical solutions of the semilinear Riesz space-fractional diffusion equations(RSFDEs)with time delay,which constitute an important class of differential equations of practical significance.We develop a novel implicit alternating direction method that can effectively and efficiently tackle the RSFDEs in both two and three dimensions.The numerical method is proved to be uniquely solvable,stable and convergent with second order accuracy in both space and time.Numerical results are presented to verify the accuracy and efficiency of the proposed numerical scheme.展开更多
In this paper,we study a new numerical technique for a class of 2D nonlinear fractional diffusion-wave equations with the Caputo-type temporal derivative and Riesz-type spatial derivative.Galerkin finite element schem...In this paper,we study a new numerical technique for a class of 2D nonlinear fractional diffusion-wave equations with the Caputo-type temporal derivative and Riesz-type spatial derivative.Galerkin finite element scheme is used for the discretization in the spatial direction,and the temporal component is discretized by a new alternating direction implicit(ADI)method.Next,we strictly prove that the numerical method is stable and convergent.Finally,to confirm our theoretical analysis,some numerical examples in 2D space are presented.展开更多
In this paper,we study splitting numerical methods for the three-dimensional Maxwell equations in the time domain.We propose a new kind of splitting finitedifference time-domain schemes on a staggered grid,which consi...In this paper,we study splitting numerical methods for the three-dimensional Maxwell equations in the time domain.We propose a new kind of splitting finitedifference time-domain schemes on a staggered grid,which consists of only two stages for each time step.It is proved by the energy method that the splitting scheme is unconditionally stable and convergent for problems with perfectly conducting boundary conditions.Both numerical dispersion analysis and numerical experiments are also presented to illustrate the efficiency of the proposed schemes.展开更多
文摘A modified alternating direction implicit algorithm is proposed to solve the full-vectorial finite-difference beam propagation method formulation based on H fields. The cross-coupling terms are neglected in the first sub-step, but evaluated and doubly used in the second sub-step. The order of two sub-steps is reversed for each transverse magnetic field component so that the cross-coupling terms are always expressed in implicit form, thus the calculation is very efficient and stable. Moreover, an improved six-point finite-difference scheme with high accuracy independent of specific structures of waveguide is also constructed to approximate the cross-coupling terms along the transverse directions. The imaginary-distance procedure is used to assess the validity and utility of the present method. The field patterns and the normalized propagation constants of the fundamental mode for a buried rectangular waveguide and a rib waveguide are presented. Solutions are in excellent agreement with the benchmark results from the modal transverse resonance method.
文摘Alternating direction implicit finite difference time domain (ADI-FDTD) method is unconditionally stable and the maximum time step is not limited by the Courant stability condition, but rather by numerical error. Compared with the conventional FDTD method, the time step of ADI-FDTD can be enlarged arbitrarily and the CPU cost can be reduced. 2D perfectly matched layer (PML) absorbing boundary condition is proposed to truncate computation space for ADI-FDTD in dispersive media using recursive convolution(RC) method and the 2D PML formulations for dispersive media are derived. ADI-FDTD formulations for dispersive media can be obtained from the simplified PML formulations. The scattering of target in dispersive soil is simulated under sine wave and Gaussian pulse excitations and numerical results of ADI-FDTD with PML are compared with FDTD. Good agreement is observed. At the same time the CPU cost for ADI-FDTD is obviously reduced.
基金supported by the Guangxi Natural Science Foundation[grant numbers 2018GXNSFBA281020,2018GXNSFAA138121]the Doctoral Starting up Foundation of Guilin University of Technology[grant number GLUTQD2016044].
文摘In this paper,two classes of Riesz space fractional partial differential equations including space-fractional and space-time-fractional ones are considered.These two models can be regarded as the generalization of the classical wave equation in two space dimensions.Combining with the Crank-Nicolson method in temporal direction,efficient alternating direction implicit Galerkin finite element methods for solving these two fractional models are developed,respectively.The corresponding stability and convergence analysis of the numerical methods are discussed.Numerical results are provided to verify the theoretical analysis.
文摘Numerical simulation of complex flow fields with multi-scale structures is one of the most important and challenging branches of computational fluid dynamics. From linear analysis and numerical experiments it has been discovered that the higher-order accurate method can give reliable and efficient computational results, as well as better resolution of the complex flow fields with multi-scale structures. Compact finite difference schemes, which feature higher-order accuracy and spectral-like resolution with smaller stencils and easier application of boundary conditions, has attracted more and more interest and attention.
基金Project supported by the National Natural Science Foundation of China(Grant No.61771407)
文摘A parabolic equation (PE) based method for analyzing composite scattering under an electromagnetic wave incidence at low grazing angle, which composes of three-dimensional (3-D) electrically large targets and rough surface, is presented and discussed. A superior high-order PE version is used to improve the accuracy at wider paraxial angles, and along with the alternating direction implicit (ADI) differential technique, the computational efficiency is further improved. The formula of bistatic normalized radar cross section is derived by definition and near-far field transformation. Numerical examples are given to show the validity and accuracy of the proposed approach, in which the results are compared with those of Kirchhoff approximation (KA) and moment of method (MoM). Furthermore, the bistatic scattering properties of composite model in which the 3-D PEC targets on or above the two-dimensional Gaussian rough surfaces under the tapered wave incidence are analyzed.
基金supported by the National Natural Science Foundation of China(No.51174236)the National Basic Research Program of China(973 Program)(No.2011CB606306)the Opening Project of State Key Laboratory of Porous Metal Materials(No.PMM-SKL-4-2012)
文摘An efficient high-order immersed interface method (IIM) is proposed to solve two-dimensional (2D) heat problems with fixed interfaces on Cartesian grids, which has the fourth-order accuracy in the maximum norm in both time and space directions. The space variable is discretized by a high-order compact (HOC) difference scheme with correction terms added at the irregular points. The time derivative is integrated by a Crank-Nicolson and alternative direction implicit (ADI) scheme. In this case, the time accuracy is just second-order. The Richardson extrapolation method is used to improve the time accuracy to fourth-order. The numerical results confirm the convergence order and the efficiency of the method.
基金The authors would like to thank the referees for their valuable comments and suggestionsThis work was supported by the National Natural Science Foundation of China,contract grant number 11271123.
文摘New numerical techniques are presented for the solution of the twodimensional time fractional evolution equation in the unit square.In these methods,Galerkin finite element is used for the spatial discretization,and,for the time stepping,new alternating direction implicit(ADI)method based on the backward Euler method combined with the first order convolution quadrature approximating the integral term are considered.The ADI Galerkin finite element method is proved to be convergent in time and in the L2 norm in space.The convergence order is O(k|ln k|+h^(r)),where k is the temporal grid size and h is spatial grid size in the x and y directions,respectively.Numerical results are presented to support our theoretical analysis.
基金supported by the National Natural Science Foundation of China grants No.11971241.
文摘Based on rectangular partition and bilinear interpolation,we construct an alternating-direction implicit(ADI)finite volume element method,which combined the merits of finite volume element method and alternating direction implicit method to solve a viscous wave equation with variable coefficients.This paper presents a general procedure to construct the alternating-direction implicit finite volume element method and gives computational schemes.Optimal error estimate in L2 norm is obtained for the schemes.Compared with the finite volume element method of the same convergence order,our method is more effective in terms of running time with the increasing of the computing scale.Numerical experiments are presented to show the efficiency of our method and numerical results are provided to support our theoretical analysis.
基金the National Natural Sciences Foundation of China
文摘An alternating direction implicit (ADI) Galerkin method with moving finite element spaces is formulated for a class of second order hyperbolic equations in two space variables. A priori H 1 error estimate is derived.
文摘In this paper, a two-dimensional nonlinear coupled Gray Scott system is simulated with a finite difference scheme and a finite volume technique. Pre and post-processing lead to a new solution called GSmFoam by understandin<span>g geometry settings and mesh information. The concentration profile chan</span>ges over time, as does the intensity of the contour patterns. The OpenFoam solver gives you the confidence to compare the pattern result with efficient numerical algorithms on the Gray Scott model.
文摘In this paper, an alternating direction Galerkin finite element method is presented for solving 2D time fractional reaction sub-diffusion equation with nonlinear source term. Firstly, one order implicit-explicit method is used for time discretization, then Galerkin finite element method is adopted for spatial discretization and obtain a fully discrete linear system. Secondly, Galerkin alternating direction procedure for the system is derived by adding an extra term. Finally, the stability and convergence of the method are analyzed rigorously. Numerical results confirm the accuracy and efficiency of the proposed method.
文摘This paper detailedly discusses the locally one-dimensional numerical methods for ef- ficiently solving the three-dimensional fractional partial differential equations, including fractional advection diffusion equation and Riesz fractional diffusion equation. The second order finite difference scheme is used to discretize the space fractional derivative and the Crank-Nicolson procedure to the time derivative. We theoretically prove and numerically verify that the presented numerical methods are unconditionally stable and second order convergent in both space and time directions. In particular, for the Riesz fractional dif- fusion equation, the idea of reducing the splitting error is used to further improve the algorithm, and the unconditional stability and convergency are also strictly proved and numerically verified for the improved scheme.
基金supported by National Natural Science Foundation(NSF)of China(Grant No.11501238)NSF of Guangdong Province(Grant No.2016A030313119)and NSF of Huizhou University(Grant No.hzu201806)+2 种基金supported by the startup fund from City University of Hong Kong and the Hong Kong RGC General Research Fund(projects Nos.12301420,12302919 and 12301218)supported by the NSF of China No.11971221the Shenzhen Sci-Tech Fund No.JCYJ20190809150413261,JCYJ20180307151603959,and JCYJ20170818153840322,and Guangdong Provincial Key Laboratory of Computational Science and Material Design(No.2019B030301001).
文摘In this paper,we consider the numerical solutions of the semilinear Riesz space-fractional diffusion equations(RSFDEs)with time delay,which constitute an important class of differential equations of practical significance.We develop a novel implicit alternating direction method that can effectively and efficiently tackle the RSFDEs in both two and three dimensions.The numerical method is proved to be uniquely solvable,stable and convergent with second order accuracy in both space and time.Numerical results are presented to verify the accuracy and efficiency of the proposed numerical scheme.
基金NSF of China[grant number:11371157]Natural Science Foundation of Anhui Higher Education Institutions of China[grant number:KJ2016A492]Natural Science Foundation of Bozhou College[grant number:BSKY201426,BSKY201535].
文摘In this paper,we study a new numerical technique for a class of 2D nonlinear fractional diffusion-wave equations with the Caputo-type temporal derivative and Riesz-type spatial derivative.Galerkin finite element scheme is used for the discretization in the spatial direction,and the temporal component is discretized by a new alternating direction implicit(ADI)method.Next,we strictly prove that the numerical method is stable and convergent.Finally,to confirm our theoretical analysis,some numerical examples in 2D space are presented.
文摘In this paper,we study splitting numerical methods for the three-dimensional Maxwell equations in the time domain.We propose a new kind of splitting finitedifference time-domain schemes on a staggered grid,which consists of only two stages for each time step.It is proved by the energy method that the splitting scheme is unconditionally stable and convergent for problems with perfectly conducting boundary conditions.Both numerical dispersion analysis and numerical experiments are also presented to illustrate the efficiency of the proposed schemes.