In this paper, the generalized nonlinear Schrodinger equation （GNLSE） is solved by an adaptive split-step Fourier method （ASSFM）. It is found that ASSFM must be used to solve GNLSE to ensure precision when the sol...In this paper, the generalized nonlinear Schrodinger equation （GNLSE） is solved by an adaptive split-step Fourier method （ASSFM）. It is found that ASSFM must be used to solve GNLSE to ensure precision when the soliton selffrequency shift is remarkable and the photonic crystal fibre （PCF） parameters vary with the frequency considerably. The precision of numerical simulation by using ASSFM is higher than that by using split-step Fourier method in the process of laser pulse propagation in PCFs due to the fact that the variation of fibre parameters with the peak frequency in the pulse spectrum can be taken into account fully.展开更多
We develop error-control based time integration algorithms for compressible fluid dynam-ics(CFD)applications and show that they are efficient and robust in both the accuracy-limited and stability-limited regime.Focusi...We develop error-control based time integration algorithms for compressible fluid dynam-ics(CFD)applications and show that they are efficient and robust in both the accuracy-limited and stability-limited regime.Focusing on discontinuous spectral element semidis-cretizations,we design new controllers for existing methods and for some new embedded Runge-Kutta pairs.We demonstrate the importance of choosing adequate controller parameters and provide a means to obtain these in practice.We compare a wide range of error-control-based methods,along with the common approach in which step size con-trol is based on the Courant-Friedrichs-Lewy(CFL)number.The optimized methods give improved performance and naturally adopt a step size close to the maximum stable CFL number at loose tolerances,while additionally providing control of the temporal error at tighter tolerances.The numerical examples include challenging industrial CFD applications.展开更多
This paper deals with the parallel diagonal implicit Runge-Kutta methods for solving DDEs with a constant delay. It is shown that the suitable choice of the predictor matrix can guarantee the stability of the methods....This paper deals with the parallel diagonal implicit Runge-Kutta methods for solving DDEs with a constant delay. It is shown that the suitable choice of the predictor matrix can guarantee the stability of the methods. It is proved that for the suitable selection of the diagonal matrix D, the method based on Radau IIA is δ-convergent, and the estimates for the non-stiff speed and the stiff speed of convergence are given.展开更多
In this paper, a split-step 0 （SST） method is introduced and used to solve the non- linear neutral stochastic differential delay equations with Poisson jumps （NSDDEwPJ）. The mean square asymptotic stability of the...In this paper, a split-step 0 （SST） method is introduced and used to solve the non- linear neutral stochastic differential delay equations with Poisson jumps （NSDDEwPJ）. The mean square asymptotic stability of the SST method for nonlinear neutral stochastic differential equations with Poisson jumps is studied. It is proved that under the one-sided Lipschitz condition and the linear growth condition, the SST method with ∈ E （0, 2 -√2） is asymptotically mean square stable for all positive step sizes, and the SST method with ∈ E （2 -√2, 1） is asymptotically mean square stable for some step sizes. It is also proved in this paper that the SST method possesses a bounded absorbing set which is independent of initial data, and the mean square dissipativity of this method is also proved.展开更多
We study a temporal step size control of explicit Runge-Kutta(RK)methods for com-pressible computational fuid dynamics(CFD),including the Navier-Stokes equations and hyperbolic systems of conservation laws such as the...We study a temporal step size control of explicit Runge-Kutta(RK)methods for com-pressible computational fuid dynamics(CFD),including the Navier-Stokes equations and hyperbolic systems of conservation laws such as the Euler equations.We demonstrate that error-based approaches are convenient in a wide range of applications and compare them to more classical step size control based on a Courant-Friedrichs-Lewy(CFL)number.Our numerical examples show that the error-based step size control is easy to use,robust,and efcient,e.g.,for(initial)transient periods,complex geometries,nonlinear shock captur-ing approaches,and schemes that use nonlinear entropy projections.We demonstrate these properties for problems ranging from well-understood academic test cases to industrially relevant large-scale computations with two disjoint code bases,the open source Julia pack-ages Trixi.jl with OrdinaryDiffEq.jl and the C/Fortran code SSDC based on PETSc.展开更多
文摘In this paper, the generalized nonlinear Schrodinger equation （GNLSE） is solved by an adaptive split-step Fourier method （ASSFM）. It is found that ASSFM must be used to solve GNLSE to ensure precision when the soliton selffrequency shift is remarkable and the photonic crystal fibre （PCF） parameters vary with the frequency considerably. The precision of numerical simulation by using ASSFM is higher than that by using split-step Fourier method in the process of laser pulse propagation in PCFs due to the fact that the variation of fibre parameters with the peak frequency in the pulse spectrum can be taken into account fully.
基金Open Access funding enabled and organized by Projekt DEAL.
文摘We develop error-control based time integration algorithms for compressible fluid dynam-ics(CFD)applications and show that they are efficient and robust in both the accuracy-limited and stability-limited regime.Focusing on discontinuous spectral element semidis-cretizations,we design new controllers for existing methods and for some new embedded Runge-Kutta pairs.We demonstrate the importance of choosing adequate controller parameters and provide a means to obtain these in practice.We compare a wide range of error-control-based methods,along with the common approach in which step size con-trol is based on the Courant-Friedrichs-Lewy(CFL)number.The optimized methods give improved performance and naturally adopt a step size close to the maximum stable CFL number at loose tolerances,while additionally providing control of the temporal error at tighter tolerances.The numerical examples include challenging industrial CFD applications.
文摘This paper deals with the parallel diagonal implicit Runge-Kutta methods for solving DDEs with a constant delay. It is shown that the suitable choice of the predictor matrix can guarantee the stability of the methods. It is proved that for the suitable selection of the diagonal matrix D, the method based on Radau IIA is δ-convergent, and the estimates for the non-stiff speed and the stiff speed of convergence are given.
文摘In this paper, a split-step 0 （SST） method is introduced and used to solve the non- linear neutral stochastic differential delay equations with Poisson jumps （NSDDEwPJ）. The mean square asymptotic stability of the SST method for nonlinear neutral stochastic differential equations with Poisson jumps is studied. It is proved that under the one-sided Lipschitz condition and the linear growth condition, the SST method with ∈ E （0, 2 -√2） is asymptotically mean square stable for all positive step sizes, and the SST method with ∈ E （2 -√2, 1） is asymptotically mean square stable for some step sizes. It is also proved in this paper that the SST method possesses a bounded absorbing set which is independent of initial data, and the mean square dissipativity of this method is also proved.
基金Open Access funding enabled and organized by Projekt DEAL.Andrew Winters was funded through Vetenskapsrådet,Sweden Grant Agreement 2020-03642 VR.Some computations were enabled by resources provided by the Swedish National Infrastructure for Computing(SNIC)at Tetralith,par-tially funded by the Swedish Research Council under Grant Agreement No.2018-05973Hugo Guillermo Castro was funded through the award P2021-0004 of King Abdullah University of Science and Technol-ogy.Some of the simulations were enabled by the Supercomputing Laboratory and the Extreme Comput-ing Research Center at King Abdullah University of Science and Technology.Gregor Gassner acknowl-edges funding through the Klaus-Tschira Stiftung via the project“HiFiLab”.Gregor Gassner and Michael Schlottke-Lakemper acknowledge funding from the Deutsche Forschungsgemeinschaft through the research unit“SNuBIC”(DFG-FOR5409).
文摘We study a temporal step size control of explicit Runge-Kutta(RK)methods for com-pressible computational fuid dynamics(CFD),including the Navier-Stokes equations and hyperbolic systems of conservation laws such as the Euler equations.We demonstrate that error-based approaches are convenient in a wide range of applications and compare them to more classical step size control based on a Courant-Friedrichs-Lewy(CFL)number.Our numerical examples show that the error-based step size control is easy to use,robust,and efcient,e.g.,for(initial)transient periods,complex geometries,nonlinear shock captur-ing approaches,and schemes that use nonlinear entropy projections.We demonstrate these properties for problems ranging from well-understood academic test cases to industrially relevant large-scale computations with two disjoint code bases,the open source Julia pack-ages Trixi.jl with OrdinaryDiffEq.jl and the C/Fortran code SSDC based on PETSc.
基金National Natural Science Foundation of China(10901074)Provincial Natural Science Foundation of Jiangxi(2008GQS0054)Postgraduate Innovation Foundation of Jiangxi Normal University(YJS2010009)
