In this paper,we establish a new algorithm to the non-overlapping Schwarz domain decomposition methods with changing transmission conditions for solving one dimensional advection reaction diffusion problem.More precis...In this paper,we establish a new algorithm to the non-overlapping Schwarz domain decomposition methods with changing transmission conditions for solving one dimensional advection reaction diffusion problem.More precisely,we first describe the new algorithm and prove the convergence results under several natural assumptions on the sequences of parameters which determine the transmission conditions.Then we give a simple method to estimate the new value of parameters in each iteration.The interesting advantage of our method is that one may update the better parameters in each iteration to save the computational cost for optimizing the parameters after many steps.Finally some numerical experiments are performed to show the behavior of the convergence rate for the new method.展开更多
The multisplitting algorithm for solving large systems of ordinary differential equations on parallel computers was introduced by Jeltsch and Pohl in [1]. On fixed time intervals conver gence results could be derived ...The multisplitting algorithm for solving large systems of ordinary differential equations on parallel computers was introduced by Jeltsch and Pohl in [1]. On fixed time intervals conver gence results could be derived if the subsystems are solving exactly.Firstly,in theis paper,we deal with an extension of the waveform relaxation algorithm by us ing multisplittin AOR method based on an overlapping block decomposition. We restricted our selves to equidistant timepoints and dealed with the case that an implicit integration method was used to solve the subsystems numerically in parallel. Then we have proved convergence of multi splitting AOR waveform relaxation algorithm on a fixed window containing a finite number of timepoints.展开更多
We present a windowing technique of waveform relaxation for dynamic systems. An effective estimation on window length is derived by an iterative error expression provided here. Relaxation processes can be speeded up i...We present a windowing technique of waveform relaxation for dynamic systems. An effective estimation on window length is derived by an iterative error expression provided here. Relaxation processes can be speeded up if one takes the windowing technique in advance. Numerical experiments are given to further illustrate the theoretical analysis.展开更多
In this paper we propose some waveform relaxation (WR) methods for solving large systems of initial value problems. Nonlinear ODEs, linear ODEs, semi-explicit DAEs and linear DAEs are discussed. The accuracy increase ...In this paper we propose some waveform relaxation (WR) methods for solving large systems of initial value problems. Nonlinear ODEs, linear ODEs, semi-explicit DAEs and linear DAEs are discussed. The accuracy increase for WR methods is investigated.展开更多
In this paper, we present an accelerated simulation approach on waveform relaxation using Krylov subspace for a large time-dependent system composed of some subsystems. This approach first allows these subsystems to b...In this paper, we present an accelerated simulation approach on waveform relaxation using Krylov subspace for a large time-dependent system composed of some subsystems. This approach first allows these subsystems to be decoupled by waveform relaxation. Then the Arnoldi procedure based on Krylov subspace is provided to accelerate the simulation of the decoupled subsystems independently. For the new approach, the convergent conditions on waveform relaxation are derived. The robust behavior is also successfully illustrated via numerical examples.展开更多
A quasi-Newton waveform relaxation （WR） algorithm for semi-linear reaction-diffusion equations is presented at first in this paper. Using the idea of energy estimate, a general proof method for convergence of the co...A quasi-Newton waveform relaxation （WR） algorithm for semi-linear reaction-diffusion equations is presented at first in this paper. Using the idea of energy estimate, a general proof method for convergence of the continuous case and the discrete case of quasi-Newton WR is given, which appears to be the superlinear rate. The semi-linear wave equation and semi-linear coupled equations can similarly be solved by quasi-Newton WR algorithm and be proved as convergent with the energy inequalities. Finally several parallel numerical experiments are implemented to confirm the effectiveness of the above theories.展开更多
In this paper,we derive and analyse waveform relaxation(WR)methods for solving differential equations evolving on a Lie-group.We present both continuous-time and discrete-time WR methods and study their convergence pr...In this paper,we derive and analyse waveform relaxation(WR)methods for solving differential equations evolving on a Lie-group.We present both continuous-time and discrete-time WR methods and study their convergence properties.In the discrete-time case,the novel methods are constructed by combining WR methods with Runge-KuttaMunthe-Kaas(RK-MK)methods.The obtained methods have both advantages of WR methods and RK-MK methods,which simplify the computation by decoupling strategy and preserve the numerical solution of Lie-group equations on a manifold.Three numerical experiments are given to illustrate the feasibility of the new WR methods.展开更多
Schwarzwaveformrelaxation(SWR)algorithmhas been investigated deeply and widely for regular time dependent problems.But for time delay problems,complete analysis of the algorithm is rare.In this paper,by using the reac...Schwarzwaveformrelaxation(SWR)algorithmhas been investigated deeply and widely for regular time dependent problems.But for time delay problems,complete analysis of the algorithm is rare.In this paper,by using the reaction diffusion equations with a constant discrete delay as the underlying model problem,we investigate the convergence behavior of the overlapping SWR algorithm with Robin transmission condition.The key point of using this transmission condition is to determine a free parameter as better as possible and it is shown that the best choice of the parameter is determined by the solution of a min-max problem,which is more complex than the one arising for regular problems without delay.We propose new notion to solve the min-max problem and obtain a quasi-optimized choice of the parameter,which is shown efficient to accelerate the convergence of the SWR algorithm.Numerical results are provided to validate the theoretical conclusions.展开更多
Nonlinear fractional differential-algebraic equations often arise in simulating integrated circuits with superconductors. How to obtain the nonnegative solutions of the equations is an important scientific problem. As...Nonlinear fractional differential-algebraic equations often arise in simulating integrated circuits with superconductors. How to obtain the nonnegative solutions of the equations is an important scientific problem. As far as we known, the nonnegativity of solutions of the nonlinear fractional differential-algebraic equations is still not studied. In this article, we investigate the nonnegativity of solutions of the equations. Firstly, we discuss the existence of nonnegative solutions of the equations, and then we show that the nonnegative solution can be approached by a monotone waveform relaxation sequence provided the initial iteration is chosen properly. The choice of initial iteration is critical and we give a method of finding it. Finally, we present an example to illustrate the efficiency of our method.展开更多
This paper is concerned with efficient numerical methods for the advectiondiffusion equation in a heterogeneous porous medium containing fractures.A dimensionally reduced fracture model is considered,in which the frac...This paper is concerned with efficient numerical methods for the advectiondiffusion equation in a heterogeneous porous medium containing fractures.A dimensionally reduced fracture model is considered,in which the fracture is represented as an interface between subdomains and is assumed to have larger permeability than the surrounding area.We develop three global-in-time domain decomposition methods coupled with operator splitting for the reduced fracture model,where the advection and the diffusion are treated separately by different numerical schemes and with different time steps.Importantly,smaller time steps can be used in the fracture-interface than in the subdomains.The first two methods are based on the physical transmission conditions,while the third one is based on the optimized Schwarz waveform relaxation approach with Ventcel-Robin transmission conditions.A discrete space-time interface system is formulated for each method and is solved iteratively and globally in time.Numerical results for two-dimensional problems with various P′eclet numbers and different types of fracture are presented to illustrate and compare the convergence and accuracy in time of the proposed methods with local time stepping.展开更多
文摘In this paper,we establish a new algorithm to the non-overlapping Schwarz domain decomposition methods with changing transmission conditions for solving one dimensional advection reaction diffusion problem.More precisely,we first describe the new algorithm and prove the convergence results under several natural assumptions on the sequences of parameters which determine the transmission conditions.Then we give a simple method to estimate the new value of parameters in each iteration.The interesting advantage of our method is that one may update the better parameters in each iteration to save the computational cost for optimizing the parameters after many steps.Finally some numerical experiments are performed to show the behavior of the convergence rate for the new method.
文摘The multisplitting algorithm for solving large systems of ordinary differential equations on parallel computers was introduced by Jeltsch and Pohl in [1]. On fixed time intervals conver gence results could be derived if the subsystems are solving exactly.Firstly,in theis paper,we deal with an extension of the waveform relaxation algorithm by us ing multisplittin AOR method based on an overlapping block decomposition. We restricted our selves to equidistant timepoints and dealed with the case that an implicit integration method was used to solve the subsystems numerically in parallel. Then we have proved convergence of multi splitting AOR waveform relaxation algorithm on a fixed window containing a finite number of timepoints.
基金Supported by the National Natural Science Foundation of China(No.60472003)the 863 Program of China(No.2001AA111042)
文摘We present a windowing technique of waveform relaxation for dynamic systems. An effective estimation on window length is derived by an iterative error expression provided here. Relaxation processes can be speeded up if one takes the windowing technique in advance. Numerical experiments are given to further illustrate the theoretical analysis.
文摘In this paper we propose some waveform relaxation (WR) methods for solving large systems of initial value problems. Nonlinear ODEs, linear ODEs, semi-explicit DAEs and linear DAEs are discussed. The accuracy increase for WR methods is investigated.
基金This work was supported by the Natural Science Foundation of China（NSFC） under grant 11071192 and the International Science and Technology Cooperation Program of China under grant 2010DFA14700.
文摘In this paper, we present an accelerated simulation approach on waveform relaxation using Krylov subspace for a large time-dependent system composed of some subsystems. This approach first allows these subsystems to be decoupled by waveform relaxation. Then the Arnoldi procedure based on Krylov subspace is provided to accelerate the simulation of the decoupled subsystems independently. For the new approach, the convergent conditions on waveform relaxation are derived. The robust behavior is also successfully illustrated via numerical examples.
基金This work was supported by the Natural Science Foundation of China （NSFC） under grant （11371287, 61663043） and Natural Science Basis Research Plan in Shaanxi Province of China under grant 2016JM5077.
文摘A quasi-Newton waveform relaxation （WR） algorithm for semi-linear reaction-diffusion equations is presented at first in this paper. Using the idea of energy estimate, a general proof method for convergence of the continuous case and the discrete case of quasi-Newton WR is given, which appears to be the superlinear rate. The semi-linear wave equation and semi-linear coupled equations can similarly be solved by quasi-Newton WR algorithm and be proved as convergent with the energy inequalities. Finally several parallel numerical experiments are implemented to confirm the effectiveness of the above theories.
基金supported by the Natural Science Foundation of China(NSFC)under grant 11871393International Science and Technology Cooperation Program of Shaanxi Key Research&Development Plan under grant 2019KWZ-08.
文摘In this paper,we derive and analyse waveform relaxation(WR)methods for solving differential equations evolving on a Lie-group.We present both continuous-time and discrete-time WR methods and study their convergence properties.In the discrete-time case,the novel methods are constructed by combining WR methods with Runge-KuttaMunthe-Kaas(RK-MK)methods.The obtained methods have both advantages of WR methods and RK-MK methods,which simplify the computation by decoupling strategy and preserve the numerical solution of Lie-group equations on a manifold.Three numerical experiments are given to illustrate the feasibility of the new WR methods.
基金supported by the NSF of China(11226312,91130003)the NSF of Sichuan University of Science and Engineering(2012XJKRL005)+1 种基金the Opening Fund of Artificial Intelligence Key Laboratory of Sichuan Province(2011RZY04)the Chinese Universities Specialized Research Fund for the Doctoral Program(20110185110020).
文摘Schwarzwaveformrelaxation(SWR)algorithmhas been investigated deeply and widely for regular time dependent problems.But for time delay problems,complete analysis of the algorithm is rare.In this paper,by using the reaction diffusion equations with a constant discrete delay as the underlying model problem,we investigate the convergence behavior of the overlapping SWR algorithm with Robin transmission condition.The key point of using this transmission condition is to determine a free parameter as better as possible and it is shown that the best choice of the parameter is determined by the solution of a min-max problem,which is more complex than the one arising for regular problems without delay.We propose new notion to solve the min-max problem and obtain a quasi-optimized choice of the parameter,which is shown efficient to accelerate the convergence of the SWR algorithm.Numerical results are provided to validate the theoretical conclusions.
基金supported by the Natural Science Foundation of China(NSFC)under grant 11501436Young Talent fund of University Association for Science and Technology in Shaanxi,China(20170701)
文摘Nonlinear fractional differential-algebraic equations often arise in simulating integrated circuits with superconductors. How to obtain the nonnegative solutions of the equations is an important scientific problem. As far as we known, the nonnegativity of solutions of the nonlinear fractional differential-algebraic equations is still not studied. In this article, we investigate the nonnegativity of solutions of the equations. Firstly, we discuss the existence of nonnegative solutions of the equations, and then we show that the nonnegative solution can be approached by a monotone waveform relaxation sequence provided the initial iteration is chosen properly. The choice of initial iteration is critical and we give a method of finding it. Finally, we present an example to illustrate the efficiency of our method.
基金partially supported by the US National Science Foundation under grant number DMS-1912626.
文摘This paper is concerned with efficient numerical methods for the advectiondiffusion equation in a heterogeneous porous medium containing fractures.A dimensionally reduced fracture model is considered,in which the fracture is represented as an interface between subdomains and is assumed to have larger permeability than the surrounding area.We develop three global-in-time domain decomposition methods coupled with operator splitting for the reduced fracture model,where the advection and the diffusion are treated separately by different numerical schemes and with different time steps.Importantly,smaller time steps can be used in the fracture-interface than in the subdomains.The first two methods are based on the physical transmission conditions,while the third one is based on the optimized Schwarz waveform relaxation approach with Ventcel-Robin transmission conditions.A discrete space-time interface system is formulated for each method and is solved iteratively and globally in time.Numerical results for two-dimensional problems with various P′eclet numbers and different types of fracture are presented to illustrate and compare the convergence and accuracy in time of the proposed methods with local time stepping.
