The local time-stepping(LTS)algorithm is an adaptive method that adjusts the time step by selecting suitable intervals for different regions based on the spatial scale of each cell and water depth and flow velocity be...The local time-stepping(LTS)algorithm is an adaptive method that adjusts the time step by selecting suitable intervals for different regions based on the spatial scale of each cell and water depth and flow velocity between cells.The method can be optimized by calculating the maximum power of two of the global time step increments in the domain,allowing the optimal time step to be approached throughout the grid.To verify the acceleration and accuracy of LTS in storm surge simulations,we developed a model to simulate astronomical storm surges along the southern coast of China.This model employs the shallow water equations as governing equations,numerical discretization using the finite volume method,and fluxes calculated by the Roe solver.By comparing the simulation results of the traditional global time-stepping algorithm with those of the LTS algorithm,we find that the latter fit the measured data better.Taking the calculation results of Typhoon Sally in 1996 as an example,we show that compared with the traditional global time-stepping algorithm,the LTS algorithm reduces computation time by 2.05 h and increases computation efficiency by 2.64 times while maintaining good accuracy.展开更多
A backward differentiation formula (BDF) has been shown to be an effective way to solve a system of ordinary differential equations (ODEs) that have some degree of stiffness. However, sometimes, due to high-frequency ...A backward differentiation formula (BDF) has been shown to be an effective way to solve a system of ordinary differential equations (ODEs) that have some degree of stiffness. However, sometimes, due to high-frequency variations in the external time series of boundary conditions, a small time-step is required to solve the ODE system throughout the entire simulation period, which can lead to a high computational cost, slower response, and need for more memory resources. One possible strategy to overcome this problem is to dynamically adjust the time-step with respect to the system’s stiffness. Therefore, small time-steps can be applied when needed, and larger time-steps can be used when allowable. This paper presents a new algorithm for adjusting the dynamic time-step based on a BDF discretization method. The parameters used to dynamically adjust the size of the time-step can be optimally specified to result in a minimum computation time and reasonable accuracy for a particular case of ODEs. The proposed algorithm was applied to solve the system of ODEs obtained from an activated sludge model (ASM) for biological wastewater treatment processes. The algorithm was tested for various solver parameters, and the optimum set of three adjustable parameters that represented minimum computation time was identified. In addition, the accuracy of the algorithm was evaluated for various sets of solver parameters.展开更多
If a traditional explicit numerical integration algorithm is used to solve motion equation in the finite element simulation of wave motion, the time-step used by numerical integration is the smallest time-step restric...If a traditional explicit numerical integration algorithm is used to solve motion equation in the finite element simulation of wave motion, the time-step used by numerical integration is the smallest time-step restricted by the stability criterion in computational region. However, the excessively small time-step is usually unnecessary for a large portion of computational region. In this paper, a varying time-step explicit numerical integration algorithm is introduced, and its basic idea is to use different time-step restricted by the stability criterion in different computational region. Finally, the feasibility of the algorithm and its effect on calculating precision are verified by numerical test.展开更多
The effect of ionospheric delay on the ground-based augmentation system under normal conditions can be mitigated by determining the value of the nominal ionospheric gradient(σvig).The nominal ionospheric gradient is ...The effect of ionospheric delay on the ground-based augmentation system under normal conditions can be mitigated by determining the value of the nominal ionospheric gradient(σvig).The nominal ionospheric gradient is generally obtained from Continuously Operating Reference Stations data by using the spatial single-difference method(mixed-pair,station-pair,or satellite-pair)or the temporal single-difference method(time-step).The time-step method uses only a single receiver,but it still contains ionospheric temporal variations.We introduce a corrected time-step method using a fixed-ionospheric pierce point from the geostationary equatorial orbit satellite and test it through simulations based on the global ionospheric model.We also investigate the effect of satellite paths on the corrected time-step method in the region of the equator,which tends to be in a more north–south direction and to have less coverage for the east–west ionospheric gradient.This study also addresses the limitations of temporal variation correction coverage and recommends using only the correction from self-observations.All processes are developed under simulations because observational data are still difficult to obtain.Our findings demonstrate that the corrected time-step method yieldsσvig values consistent with other approaches.展开更多
Forward modeling of elastic wave propagation in porous media has great importance for understanding and interpreting the influences of rock properties on characteristics of seismic wavefield. However,the finite-differ...Forward modeling of elastic wave propagation in porous media has great importance for understanding and interpreting the influences of rock properties on characteristics of seismic wavefield. However,the finite-difference forward-modeling method is usually implemented with global spatial grid-size and time-step; it consumes large amounts of computational cost when small-scaled oil/gas-bearing structures or large velocity-contrast exist underground. To overcome this handicap,combined with variable grid-size and time-step,this paper developed a staggered-grid finite-difference scheme for elastic wave modeling in porous media. Variable finite-difference coefficients and wavefield interpolation were used to realize the transition of wave propagation between regions of different grid-size. The accuracy and efficiency of the algorithm were shown by numerical examples. The proposed method is advanced with low computational cost in elastic wave simulation for heterogeneous oil/gas reservoirs.展开更多
There are two models in use today to analyze structural responses when subjected to earthquake ground motions, the Displacement Input Model （DIM） and the Acceleration Input Model （AIM）. The time steps used in dire...There are two models in use today to analyze structural responses when subjected to earthquake ground motions, the Displacement Input Model （DIM） and the Acceleration Input Model （AIM）. The time steps used in direct integration methods for these models are analyzed to examine the suitability of DIM. Numerical results are presented and show that the time-step for DIM is about the same as for AIM, and achieves the same accuracy. This is contrary to previous research that reported that there are several sources of numerical errors associated with the direct application of earthquake displacement loading, and a very small time step is required to define the displacement record and to integrate the dynamic equilibrium equation. It is shown in this paper that DIM is as accurate and suitable as, if not more than, AIM for analyzing the response of a structure to uniformly distributed and spatially varying ground motions.展开更多
The numerical time step integrations of PDEs are mainly carried out by the finitedifference method to date. However,when the time step becomes longer, it causes theproblem of numerical instability,. The explicit integ...The numerical time step integrations of PDEs are mainly carried out by the finitedifference method to date. However,when the time step becomes longer, it causes theproblem of numerical instability,. The explicit integration schemes derived by the singlepoint precise integration method given in this paper are proved unconditionally stable.Comparisons between the schemes derived by the finite difference method and theschemes by the method employed in the present paper are made for diffusion andconvective-diffusion equations. Nunierical examples show the superiority of the singlepoint integration method.展开更多
The recently developed DOC kernels technique has been successful in the stability and convergence analysis for variable time-step BDF2 schemes.However,it may not be readily applicable to problems exhibiting an initial...The recently developed DOC kernels technique has been successful in the stability and convergence analysis for variable time-step BDF2 schemes.However,it may not be readily applicable to problems exhibiting an initial singularity.In the numerical simulations of solutions with initial singularity,variable time-step schemes like the graded mesh are always adopted to achieve the optimal convergence,whose first adjacent time-step ratio may become pretty large so that the acquired restriction is not satisfied.In this paper,we revisit the variable time-step implicit-explicit two-step backward differentiation formula(IMEX BDF2)scheme to solve the parabolic integro-differential equations(PIDEs)with initial singularity.We obtain the sharp error estimate under a mild restriction condition of adjacent time-step ratios r_(k):=T_(k)/T_(k-1)<r_(max)=4.8645(k≥3)and a much mild requirement on the first ratio,i.e.r_(2)>0.This leads to the validation of our analysis of the variable time-step IMEX BDF2 scheme when the initial singularity is dealt by a simple strategy,i.e.the graded mesh t_(k)=T(k/N)^(γ).In this situation,the convergence order of O(N^(-min(2,γα))is achieved,where N denotes the total number of mesh points andαindicates the regularity of the exact solution.This is,the optimal convergence will be achieved by taking%γ_(opt)=2/α.Numerical examples are provided to demonstrate our theoretical analysis.展开更多
To study the influence of skewed rotors and different skew angles on the losses of squirrel cage asynchronous motors,a 5.5-kW motor was taken as an example and the multi-sliced field-circuit coupled time stepping fini...To study the influence of skewed rotors and different skew angles on the losses of squirrel cage asynchronous motors,a 5.5-kW motor was taken as an example and the multi-sliced field-circuit coupled time stepping finite element method(T-S FEM)was used to analyze the axially non-uniform fundamental and harmonic field distribution characteristics at typical locations in the stator and rotor cores.The major conclusions are:firstly the skewed rotor exhibits a decrease in the harmonic copper losses caused by slot harmonic currents in the stator winding and rotor bars.Secondly,the skewed rotor shifts the non-uniform distribution of field in the axial direction,which leads to more severe saturation and an increase in iron losses.The heavier the load,the more pronounced the increase in iron losses.Furthermore,the influences of different skew angles on motor losses are studied systematically,with skew angles from 0.5 to 1.5 stator tooth pitch.It is found that the lowest total loss occurs at 0.8 stator tooth pitch,and the slot harmonics can be decreased effectively.展开更多
This paper studies the numerical simulations for the Cahn-Hilliard equation which describes a phase separation phenomenon.The numerical simulation of the Cahn-Hilliardmodel needs very long time to reach the steady sta...This paper studies the numerical simulations for the Cahn-Hilliard equation which describes a phase separation phenomenon.The numerical simulation of the Cahn-Hilliardmodel needs very long time to reach the steady state,and therefore large time-stepping methods become useful.The main objective of this work is to construct the unconditionally energy stable finite difference scheme so that the large time steps can be used in the numerical simulations.The equation is discretized by the central difference scheme in space and fully implicit second-order scheme in time.The proposed scheme is proved to be unconditionally energy stable and mass-conservative.An error estimate for the numerical solution is also obtained with second order in both space and time.By using this energy stable scheme,an adaptive time-stepping strategy is proposed,which selects time steps adaptively based on the variation of the free energy against time.The numerical experiments are presented to demonstrate the effectiveness of the adaptive time-stepping approach.展开更多
In this paper,we will develop a first order and a second order convex splitting,and a first order linear energy stable fully discrete local discontinuous Galerkin(LDG)methods for the modified phase field crystal(MPFC)...In this paper,we will develop a first order and a second order convex splitting,and a first order linear energy stable fully discrete local discontinuous Galerkin(LDG)methods for the modified phase field crystal(MPFC)equation.In which,the first order linear scheme is based on the invariant energy quadratization approach.The MPFC equation is a damped wave equation,and to preserve an energy stability,it is necessary to introduce a pseudo energy,which all increase the difficulty of constructing numerical methods comparing with the phase field crystal(PFC)equation.Due to the severe time step restriction of explicit timemarchingmethods,we introduce the first order and second order semi-implicit schemes,which are proved to be unconditionally energy stable.In order to improve the temporal accuracy,the semi-implicit spectral deferred correction(SDC)method combining with the first order convex splitting scheme is employed.Numerical simulations of the MPFC equation always need long time to reach steady state,and then adaptive time-stepping method is necessary and of paramount importance.The schemes at the implicit time level are linear or nonlinear and we solve them by multigrid solver.Numerical experiments of the accuracy and long time simulations are presented demonstrating the capability and efficiency of the proposed methods,and the effectiveness of the adaptive time-stepping strategy.展开更多
The main purpose of this work is to contrast and analyze a large timestepping numerical method for the Swift-Hohenberg(SH)equation.This model requires very large time simulation to reach steady state,so developing a l...The main purpose of this work is to contrast and analyze a large timestepping numerical method for the Swift-Hohenberg(SH)equation.This model requires very large time simulation to reach steady state,so developing a large time step algorithm becomes necessary to improve the computational efficiency.In this paper,a semi-implicit Euler schemes in time is adopted.An extra artificial term is added to the discretized system in order to preserve the energy stability unconditionally.The stability property is proved rigorously based on an energy approach.Numerical experiments are used to demonstrate the effectiveness of the large time-stepping approaches by comparing with the classical scheme.展开更多
基金National Natural Science Foundation of China(No.52071306)the Natural Science Foundation of Shandong Province(No.ZR2019MEE050)the Natural Science Foundation of Zhejiang Province(No.LZ22E090003).
文摘The local time-stepping(LTS)algorithm is an adaptive method that adjusts the time step by selecting suitable intervals for different regions based on the spatial scale of each cell and water depth and flow velocity between cells.The method can be optimized by calculating the maximum power of two of the global time step increments in the domain,allowing the optimal time step to be approached throughout the grid.To verify the acceleration and accuracy of LTS in storm surge simulations,we developed a model to simulate astronomical storm surges along the southern coast of China.This model employs the shallow water equations as governing equations,numerical discretization using the finite volume method,and fluxes calculated by the Roe solver.By comparing the simulation results of the traditional global time-stepping algorithm with those of the LTS algorithm,we find that the latter fit the measured data better.Taking the calculation results of Typhoon Sally in 1996 as an example,we show that compared with the traditional global time-stepping algorithm,the LTS algorithm reduces computation time by 2.05 h and increases computation efficiency by 2.64 times while maintaining good accuracy.
文摘A backward differentiation formula (BDF) has been shown to be an effective way to solve a system of ordinary differential equations (ODEs) that have some degree of stiffness. However, sometimes, due to high-frequency variations in the external time series of boundary conditions, a small time-step is required to solve the ODE system throughout the entire simulation period, which can lead to a high computational cost, slower response, and need for more memory resources. One possible strategy to overcome this problem is to dynamically adjust the time-step with respect to the system’s stiffness. Therefore, small time-steps can be applied when needed, and larger time-steps can be used when allowable. This paper presents a new algorithm for adjusting the dynamic time-step based on a BDF discretization method. The parameters used to dynamically adjust the size of the time-step can be optimally specified to result in a minimum computation time and reasonable accuracy for a particular case of ODEs. The proposed algorithm was applied to solve the system of ODEs obtained from an activated sludge model (ASM) for biological wastewater treatment processes. The algorithm was tested for various solver parameters, and the optimum set of three adjustable parameters that represented minimum computation time was identified. In addition, the accuracy of the algorithm was evaluated for various sets of solver parameters.
基金National Natural Science Foundation of China (50178065), 973 Program (2002CB412706), National Social Com-monweal Research Foundation (2002DIB30076) and Joint Seismological Science Foundation (101066).
文摘If a traditional explicit numerical integration algorithm is used to solve motion equation in the finite element simulation of wave motion, the time-step used by numerical integration is the smallest time-step restricted by the stability criterion in computational region. However, the excessively small time-step is usually unnecessary for a large portion of computational region. In this paper, a varying time-step explicit numerical integration algorithm is introduced, and its basic idea is to use different time-step restricted by the stability criterion in different computational region. Finally, the feasibility of the algorithm and its effect on calculating precision are verified by numerical test.
基金funding from BRIN through the Research Collaboration Program with ORPA(No.2/III.1/HK/2024)Prayitno Abadi is participating in this study as part of a Memorandum of Understanding for Research Collaboration on Regional Ionospheric Observation at Telkom University(No.092/SAM3/TE-DEK/2021).
文摘The effect of ionospheric delay on the ground-based augmentation system under normal conditions can be mitigated by determining the value of the nominal ionospheric gradient(σvig).The nominal ionospheric gradient is generally obtained from Continuously Operating Reference Stations data by using the spatial single-difference method(mixed-pair,station-pair,or satellite-pair)or the temporal single-difference method(time-step).The time-step method uses only a single receiver,but it still contains ionospheric temporal variations.We introduce a corrected time-step method using a fixed-ionospheric pierce point from the geostationary equatorial orbit satellite and test it through simulations based on the global ionospheric model.We also investigate the effect of satellite paths on the corrected time-step method in the region of the equator,which tends to be in a more north–south direction and to have less coverage for the east–west ionospheric gradient.This study also addresses the limitations of temporal variation correction coverage and recommends using only the correction from self-observations.All processes are developed under simulations because observational data are still difficult to obtain.Our findings demonstrate that the corrected time-step method yieldsσvig values consistent with other approaches.
基金supported by the National Basic Research Program of China (No. 2013CB228604)the National Science and Technology Major Project (No. 2011ZX05030-004-002,2011ZX05019-003)the National Natural Science Foundation (No. 41004050)
文摘Forward modeling of elastic wave propagation in porous media has great importance for understanding and interpreting the influences of rock properties on characteristics of seismic wavefield. However,the finite-difference forward-modeling method is usually implemented with global spatial grid-size and time-step; it consumes large amounts of computational cost when small-scaled oil/gas-bearing structures or large velocity-contrast exist underground. To overcome this handicap,combined with variable grid-size and time-step,this paper developed a staggered-grid finite-difference scheme for elastic wave modeling in porous media. Variable finite-difference coefficients and wavefield interpolation were used to realize the transition of wave propagation between regions of different grid-size. The accuracy and efficiency of the algorithm were shown by numerical examples. The proposed method is advanced with low computational cost in elastic wave simulation for heterogeneous oil/gas reservoirs.
文摘There are two models in use today to analyze structural responses when subjected to earthquake ground motions, the Displacement Input Model （DIM） and the Acceleration Input Model （AIM）. The time steps used in direct integration methods for these models are analyzed to examine the suitability of DIM. Numerical results are presented and show that the time-step for DIM is about the same as for AIM, and achieves the same accuracy. This is contrary to previous research that reported that there are several sources of numerical errors associated with the direct application of earthquake displacement loading, and a very small time step is required to define the displacement record and to integrate the dynamic equilibrium equation. It is shown in this paper that DIM is as accurate and suitable as, if not more than, AIM for analyzing the response of a structure to uniformly distributed and spatially varying ground motions.
文摘The numerical time step integrations of PDEs are mainly carried out by the finitedifference method to date. However,when the time step becomes longer, it causes theproblem of numerical instability,. The explicit integration schemes derived by the singlepoint precise integration method given in this paper are proved unconditionally stable.Comparisons between the schemes derived by the finite difference method and theschemes by the method employed in the present paper are made for diffusion andconvective-diffusion equations. Nunierical examples show the superiority of the singlepoint integration method.
基金supported by the NSFC(Grant No.12171376)the Fundamental Research Funds for the Central Universities(Grant No.2042021kf0050)and WHU-2022-SYJS-0002+1 种基金Yana Di is partially supported by the NSFC(Grant Nos.12271048,12171042)the Guangdong Key Laboratory(Grant No.2022B1212010006).
文摘The recently developed DOC kernels technique has been successful in the stability and convergence analysis for variable time-step BDF2 schemes.However,it may not be readily applicable to problems exhibiting an initial singularity.In the numerical simulations of solutions with initial singularity,variable time-step schemes like the graded mesh are always adopted to achieve the optimal convergence,whose first adjacent time-step ratio may become pretty large so that the acquired restriction is not satisfied.In this paper,we revisit the variable time-step implicit-explicit two-step backward differentiation formula(IMEX BDF2)scheme to solve the parabolic integro-differential equations(PIDEs)with initial singularity.We obtain the sharp error estimate under a mild restriction condition of adjacent time-step ratios r_(k):=T_(k)/T_(k-1)<r_(max)=4.8645(k≥3)and a much mild requirement on the first ratio,i.e.r_(2)>0.This leads to the validation of our analysis of the variable time-step IMEX BDF2 scheme when the initial singularity is dealt by a simple strategy,i.e.the graded mesh t_(k)=T(k/N)^(γ).In this situation,the convergence order of O(N^(-min(2,γα))is achieved,where N denotes the total number of mesh points andαindicates the regularity of the exact solution.This is,the optimal convergence will be achieved by taking%γ_(opt)=2/α.Numerical examples are provided to demonstrate our theoretical analysis.
基金supported by the National High Technology Research and Development Program of China("863"Program)(Grant No.2009AA05Z207)
文摘To study the influence of skewed rotors and different skew angles on the losses of squirrel cage asynchronous motors,a 5.5-kW motor was taken as an example and the multi-sliced field-circuit coupled time stepping finite element method(T-S FEM)was used to analyze the axially non-uniform fundamental and harmonic field distribution characteristics at typical locations in the stator and rotor cores.The major conclusions are:firstly the skewed rotor exhibits a decrease in the harmonic copper losses caused by slot harmonic currents in the stator winding and rotor bars.Secondly,the skewed rotor shifts the non-uniform distribution of field in the axial direction,which leads to more severe saturation and an increase in iron losses.The heavier the load,the more pronounced the increase in iron losses.Furthermore,the influences of different skew angles on motor losses are studied systematically,with skew angles from 0.5 to 1.5 stator tooth pitch.It is found that the lowest total loss occurs at 0.8 stator tooth pitch,and the slot harmonics can be decreased effectively.
基金We would like to thank Prof.Houde Han of Tsinghua University and Prof.Qiang Du of Penn State University for their helpful discussions.Z.R.Zhang was supported by National NSF of China under Grant 10601007Z.H.Qiao was supported by the FRG grants of the Hong Kong Baptist University under Grant No.FRG2/09-10/034.
文摘This paper studies the numerical simulations for the Cahn-Hilliard equation which describes a phase separation phenomenon.The numerical simulation of the Cahn-Hilliardmodel needs very long time to reach the steady state,and therefore large time-stepping methods become useful.The main objective of this work is to construct the unconditionally energy stable finite difference scheme so that the large time steps can be used in the numerical simulations.The equation is discretized by the central difference scheme in space and fully implicit second-order scheme in time.The proposed scheme is proved to be unconditionally energy stable and mass-conservative.An error estimate for the numerical solution is also obtained with second order in both space and time.By using this energy stable scheme,an adaptive time-stepping strategy is proposed,which selects time steps adaptively based on the variation of the free energy against time.The numerical experiments are presented to demonstrate the effectiveness of the adaptive time-stepping approach.
基金Research of R.Guo is supported by NSFC grant No.11601490Research of Y.Xu is supported by NSFC grant No.11371342,11626253,91630207.
文摘In this paper,we will develop a first order and a second order convex splitting,and a first order linear energy stable fully discrete local discontinuous Galerkin(LDG)methods for the modified phase field crystal(MPFC)equation.In which,the first order linear scheme is based on the invariant energy quadratization approach.The MPFC equation is a damped wave equation,and to preserve an energy stability,it is necessary to introduce a pseudo energy,which all increase the difficulty of constructing numerical methods comparing with the phase field crystal(PFC)equation.Due to the severe time step restriction of explicit timemarchingmethods,we introduce the first order and second order semi-implicit schemes,which are proved to be unconditionally energy stable.In order to improve the temporal accuracy,the semi-implicit spectral deferred correction(SDC)method combining with the first order convex splitting scheme is employed.Numerical simulations of the MPFC equation always need long time to reach steady state,and then adaptive time-stepping method is necessary and of paramount importance.The schemes at the implicit time level are linear or nonlinear and we solve them by multigrid solver.Numerical experiments of the accuracy and long time simulations are presented demonstrating the capability and efficiency of the proposed methods,and the effectiveness of the adaptive time-stepping strategy.
基金supported by the Fundamental Research Funds for the CentralUniversities andNationalNSF of China under grantNos.11271048,1130021 and 11571054.
文摘The main purpose of this work is to contrast and analyze a large timestepping numerical method for the Swift-Hohenberg(SH)equation.This model requires very large time simulation to reach steady state,so developing a large time step algorithm becomes necessary to improve the computational efficiency.In this paper,a semi-implicit Euler schemes in time is adopted.An extra artificial term is added to the discretized system in order to preserve the energy stability unconditionally.The stability property is proved rigorously based on an energy approach.Numerical experiments are used to demonstrate the effectiveness of the large time-stepping approaches by comparing with the classical scheme.
