This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either...This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either finite difference(FD)or local discontinuous Galerkin(DG)spatial discretization.We analyze the stability of the fully discrete scheme,on a uniform mesh with periodic boundary conditions,using the Fourier method.For the linearized KdV equation,the IMEX schemes are stable under the standard Courant-Friedrichs-Lewy(CFL)conditionτ≤λh.Here,λis the CFL number,τis the time-step size,and h is the spatial mesh size.We study several IMEX schemes and characterize their CFL number as a function ofθ=d/h^(2)with d being the dispersion coefficient,which leads to several interesting observations.We also investigate the asymptotic behaviors of the CFL number for sufficiently refined meshes and derive the necessary conditions for the asymptotic stability of the IMEX-RK methods.Some numerical experiments are provided in the paper to illustrate the performance of IMEX methods under different time-step constraints.展开更多
Efficient and accurate simulation of unsteady flow presents a significant challenge that needs to be overcome in computational fluid dynamics.Temporal discretization method plays a crucial role in the simulation of un...Efficient and accurate simulation of unsteady flow presents a significant challenge that needs to be overcome in computational fluid dynamics.Temporal discretization method plays a crucial role in the simulation of unsteady flows.To enhance computational efficiency,we propose the Implicit-Explicit Two-Step Runge-Kutta(IMEX-TSRK)time-stepping discretization methods for unsteady flows,and develop a novel adaptive algorithm that correctly partitions spatial regions to apply implicit or explicit methods.The novel adaptive IMEX-TSRK schemes effectively handle the numerical stiffness of the small grid size and improve computational efficiency.Compared to implicit and explicit Runge-Kutta(RK)schemes,the IMEX-TSRK methods achieve the same order of accuracy with fewer first derivative calculations.Numerical case tests demonstrate that the IMEX-TSRK methods maintain numerical stability while enhancing computational efficiency.Specifically,in high Reynolds number flows,the computational efficiency of the IMEX-TSRK methods surpasses that of explicit RK schemes by more than one order of magnitude,and that of implicit RK schemes several times over.展开更多
In this paper,a fully discrete stability analysis is carried out for the direct discontinuous Galerkin(DDG)methods coupled with Runge-Kutta-type implicit-explicit time marching,for solving one-dimensional linear conve...In this paper,a fully discrete stability analysis is carried out for the direct discontinuous Galerkin(DDG)methods coupled with Runge-Kutta-type implicit-explicit time marching,for solving one-dimensional linear convection-diffusion problems.In the spatial discretization,both the original DDG methods and the refined DDG methods with interface corrections are considered.In the time discretization,the convection term is treated explicitly and the diffusion term implicitly.By the energy method,we show that the corresponding fully discrete schemes are unconditionally stable,in the sense that the time-stepis only required to be upper bounded by a constant which is independent of the mesh size h.Opti-mal error estimate is also obtained by the aid of a special global projection.Numerical experiments are given to verify the stability and accuracy of the proposed schemes.展开更多
This paper considers pricing European options under the well-known of SVJ model of Bates and related computational methods. According to the no-arbitrage principle, we first derive a partial differential equation that...This paper considers pricing European options under the well-known of SVJ model of Bates and related computational methods. According to the no-arbitrage principle, we first derive a partial differential equation that the value of any European contingent claim should satisfy, where the asset price obeys the SVJ model. This equation is numerically solved by using the implicit- explicit backward difference method and time semi-discretization. In order to explain the validity of our method, the stability of time semi-discretization scheme is also proved. Finally, we use a simulation example to illustrate the efficiency of the method.展开更多
We consider the construction of semi-implicit linear multistep methods that can be applied to time-dependent PDEs where the separation of scales in additive form,typically used in implicit-explicit(IMEX)methods,is not...We consider the construction of semi-implicit linear multistep methods that can be applied to time-dependent PDEs where the separation of scales in additive form,typically used in implicit-explicit(IMEX)methods,is not possible.As shown in Boscarino et al.(J.Sci.Comput.68:975-1001,2016)for Runge-Kutta methods,these semi-implicit techniques give a great flexibility,and allow,in many cases,the construction of simple linearly implicit schemes with no need of iterative solvers.In this work,we develop a general setting for the construction of high order semi-implicit linear multistep methods and analyze their stability properties for a prototype lineal'advection-diffusion equation and in the setting of strong stability preserving(SSP)methods.Our findings are demonstrated on several examples,including nonlinear reaction-diffusion and convection-diffusion problems.展开更多
In this paper,two fully-discrete local discontinuous Galerkin(LDG)methods are applied to the growth-mediated autochemotactic pattern formation model in self-propelling bacteria.The numerical methods are linear and dec...In this paper,two fully-discrete local discontinuous Galerkin(LDG)methods are applied to the growth-mediated autochemotactic pattern formation model in self-propelling bacteria.The numerical methods are linear and decoupled,which greatly improve the computational efficiency.In order to resolve the time level mismatch of the discretization process,a special time marching method with high-order accuracy is constructed.Under the condition of slight time step constraints,the optimal error estimates of this method are given.Moreover,the theoretical results are verified by numerical experiments.Real simulations show the patterns of spots,rings,stripes as well as inverted spots because of the interplay of chemotactic drift and growth rate of the cells.展开更多
Implicit-explicit （IMEX） linear multistep methods are popular techniques for solving partial differential equations （PDEs） with terms of different types. While fixed timestep versions of such schemes have been dev...Implicit-explicit （IMEX） linear multistep methods are popular techniques for solving partial differential equations （PDEs） with terms of different types. While fixed timestep versions of such schemes have been developed and studied, implicit-explicit schemes also naturally arise in general situations where the temporal smoothness of the solution changes. In this paper we consider easily implementable variable step-size implicit-explicit （VSIMEX） linear multistep methods for time-dependent PDEs. Families of order-p, pstep VSIMEX schemes are constructed and analyzed, where p ranges from 1 to 4. The corresponding schemes are simple to implement and have the property that they reduce to the classical IMEX schemes whenever constant time step-sizes are imposed. The methods are validated on the Burgers＇ equation. These results demonstrate that by varying the time step-size, VSIMEX methods can outperform their fixed time step counterparts while still maintaining good numerical behavior.展开更多
Implicit-explicit Runge-Kutta-Rosenbrock methods are proposed to solve nonlinear sti ordinary di erential equations by combining linearly implicit Rosenbrock methods with explicit Runge-Kutta methods.First,the general...Implicit-explicit Runge-Kutta-Rosenbrock methods are proposed to solve nonlinear sti ordinary di erential equations by combining linearly implicit Rosenbrock methods with explicit Runge-Kutta methods.First,the general order conditions up to order 3 are obtained.Then,for the nonlinear sti initial-value problems satisfying the one-sided Lipschitz condition and a class of singularly perturbed initial-value problems,the corresponding errors of the implicit-explicit methods are analysed.At last,some numerical examples are given to verify the validity of the obtained theoretical results and the e ectiveness of the methods.展开更多
This work develops a fully discrete implicit-explicit finite element scheme for a parabolicordinary system with a nonlinear reaction term which is known as the FitzHugh-Nagumo model from physiology.The first-order bac...This work develops a fully discrete implicit-explicit finite element scheme for a parabolicordinary system with a nonlinear reaction term which is known as the FitzHugh-Nagumo model from physiology.The first-order backward Euler discretization for the time derivative,and an implicit-explicit discretization for the nonlinear reaction term are employed for the model,with a simple linearization technique used to make the process of solving equations more efficient.The stability and convergence of the fully discrete implicit-explicit finite element method are proved,which shows that the FitzHugh-Nagumo model is accurately solved and the trajectory of potential transmission is obtained.The numerical results are also reported to verify the convergence results and the st ability of the proposed method.展开更多
We consider the drift-diffusion （DD） model of one dimensional semiconductor devices, which is a system involving not only first derivative convection terms but also second derivative diffusion terms and a coupled Po...We consider the drift-diffusion （DD） model of one dimensional semiconductor devices, which is a system involving not only first derivative convection terms but also second derivative diffusion terms and a coupled Poisson potential equation. Optimal error estimates are obtained for both the semi-discrete and fully discrete local discontinuous Galerkin （LDG） schemes with smooth solutions. In the fully discrete scheme, we couple the implicit-explicit （IMEX） time discretization with the LDG spatial diseretization, in order to allow larger time steps and to save computational cost. The main technical difficulty in the analysis is to treat the inter-element jump terms which arise from the discontinuous nature of the numerical method and the nonlinearity and coupling of the models. A simulation is also performed to validate the analysis.展开更多
In this paper,we present a mesh adaptation algorithm for the unsteady compressible Navier-Stokes equations under the framework of local discontinuous Galerkin methods coupled with implicit-explicit Runge-Kutta or spec...In this paper,we present a mesh adaptation algorithm for the unsteady compressible Navier-Stokes equations under the framework of local discontinuous Galerkin methods coupled with implicit-explicit Runge-Kutta or spectral deferred correction time discretization methods.In both of the two high order semi-implicit time integration methods,the convective flux is treated explicitly and the viscous and heat fluxes are treated implicitly.The remarkable benefits of such semi-implicit temporal discretizations are that they can not only overcome the stringent time step restriction compared with time explicit methods,but also avoid the construction of the large Jacobian matrix as is done for fully implicit methods,thus are relatively easy to implement.To save computing time as well as capture the flow structures of interest accurately,a local mesh refinement(h-adaptive)technique,in which we present detailed criteria for selecting candidate elements and complete strategies to refine and coarsen them,is also applied for the Navier-Stokes equations.Numerical experiments are provided to illustrate the high order accuracy,efficiency and capabilities of the semi-implicit schemes in combination with adaptive local discontinuous Galerkin methods for the Navier-Stokes equations.展开更多
Inspiral of binary black holes occurs over a time-scale of many orbits,far longer than the dynamical time-scale of the individual black holes.Explicit evolutions of a binary system therefore require excessively many t...Inspiral of binary black holes occurs over a time-scale of many orbits,far longer than the dynamical time-scale of the individual black holes.Explicit evolutions of a binary system therefore require excessively many time-steps to capture interesting dynamics.We present a strategy to overcome the Courant-Friedrichs-Lewy condition in such evolutions,one relying on modern implicit-explicit ODE solvers and multidomain spectral methods for elliptic equations.Our analysis considers the model problem of a forced scalar field propagating on a generic curved background.Nevertheless,we encounter and address a number of issues pertinent to the binary black hole problem in full general relativity.Specializing to the Schwarzschild geometry in KerrSchild coordinates,we document the results of several numerical experiments testing our strategy.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
基金supported by the NSF under Grant DMS-2208391sponsored by the NSF under Grant DMS-1753581.
文摘This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either finite difference(FD)or local discontinuous Galerkin(DG)spatial discretization.We analyze the stability of the fully discrete scheme,on a uniform mesh with periodic boundary conditions,using the Fourier method.For the linearized KdV equation,the IMEX schemes are stable under the standard Courant-Friedrichs-Lewy(CFL)conditionτ≤λh.Here,λis the CFL number,τis the time-step size,and h is the spatial mesh size.We study several IMEX schemes and characterize their CFL number as a function ofθ=d/h^(2)with d being the dispersion coefficient,which leads to several interesting observations.We also investigate the asymptotic behaviors of the CFL number for sufficiently refined meshes and derive the necessary conditions for the asymptotic stability of the IMEX-RK methods.Some numerical experiments are provided in the paper to illustrate the performance of IMEX methods under different time-step constraints.
基金supported by the National Natural Science Foundation of China(No.92252201)the Fundamental Research Funds for the Central Universitiesthe Academic Excellence Foundation of Beihang University(BUAA)for PhD Students。
文摘Efficient and accurate simulation of unsteady flow presents a significant challenge that needs to be overcome in computational fluid dynamics.Temporal discretization method plays a crucial role in the simulation of unsteady flows.To enhance computational efficiency,we propose the Implicit-Explicit Two-Step Runge-Kutta(IMEX-TSRK)time-stepping discretization methods for unsteady flows,and develop a novel adaptive algorithm that correctly partitions spatial regions to apply implicit or explicit methods.The novel adaptive IMEX-TSRK schemes effectively handle the numerical stiffness of the small grid size and improve computational efficiency.Compared to implicit and explicit Runge-Kutta(RK)schemes,the IMEX-TSRK methods achieve the same order of accuracy with fewer first derivative calculations.Numerical case tests demonstrate that the IMEX-TSRK methods maintain numerical stability while enhancing computational efficiency.Specifically,in high Reynolds number flows,the computational efficiency of the IMEX-TSRK methods surpasses that of explicit RK schemes by more than one order of magnitude,and that of implicit RK schemes several times over.
基金the NSFC grant 11871428the Nature Science Research Program for Colleges and Universities of Jiangsu Province grant 20KJB110011Qiang Zhang:Research supported by the NSFC grant 11671199。
文摘In this paper,a fully discrete stability analysis is carried out for the direct discontinuous Galerkin(DDG)methods coupled with Runge-Kutta-type implicit-explicit time marching,for solving one-dimensional linear convection-diffusion problems.In the spatial discretization,both the original DDG methods and the refined DDG methods with interface corrections are considered.In the time discretization,the convection term is treated explicitly and the diffusion term implicitly.By the energy method,we show that the corresponding fully discrete schemes are unconditionally stable,in the sense that the time-stepis only required to be upper bounded by a constant which is independent of the mesh size h.Opti-mal error estimate is also obtained by the aid of a special global projection.Numerical experiments are given to verify the stability and accuracy of the proposed schemes.
文摘This paper considers pricing European options under the well-known of SVJ model of Bates and related computational methods. According to the no-arbitrage principle, we first derive a partial differential equation that the value of any European contingent claim should satisfy, where the asset price obeys the SVJ model. This equation is numerically solved by using the implicit- explicit backward difference method and time semi-discretization. In order to explain the validity of our method, the stability of time semi-discretization scheme is also proved. Finally, we use a simulation example to illustrate the efficiency of the method.
基金Open Access funding provided by Universita degli Studi di Verona.
文摘We consider the construction of semi-implicit linear multistep methods that can be applied to time-dependent PDEs where the separation of scales in additive form,typically used in implicit-explicit(IMEX)methods,is not possible.As shown in Boscarino et al.(J.Sci.Comput.68:975-1001,2016)for Runge-Kutta methods,these semi-implicit techniques give a great flexibility,and allow,in many cases,the construction of simple linearly implicit schemes with no need of iterative solvers.In this work,we develop a general setting for the construction of high order semi-implicit linear multistep methods and analyze their stability properties for a prototype lineal'advection-diffusion equation and in the setting of strong stability preserving(SSP)methods.Our findings are demonstrated on several examples,including nonlinear reaction-diffusion and convection-diffusion problems.
基金supported by National Natural Science Foundation of China(Grant No.11801569)Natural Science Foundation of Shandong Province(CN)(Grant No.ZR2021MA001)the Fundamental Research Funds for the Central Universities(Grant Nos.22CX03025A and 22CX03020A).
文摘In this paper,two fully-discrete local discontinuous Galerkin(LDG)methods are applied to the growth-mediated autochemotactic pattern formation model in self-propelling bacteria.The numerical methods are linear and decoupled,which greatly improve the computational efficiency.In order to resolve the time level mismatch of the discretization process,a special time marching method with high-order accuracy is constructed.Under the condition of slight time step constraints,the optimal error estimates of this method are given.Moreover,the theoretical results are verified by numerical experiments.Real simulations show the patterns of spots,rings,stripes as well as inverted spots because of the interplay of chemotactic drift and growth rate of the cells.
基金supported by an NSERC Canada Postgraduate Scholarshipsupported by a grant from NSERC Canada
文摘Implicit-explicit （IMEX） linear multistep methods are popular techniques for solving partial differential equations （PDEs） with terms of different types. While fixed timestep versions of such schemes have been developed and studied, implicit-explicit schemes also naturally arise in general situations where the temporal smoothness of the solution changes. In this paper we consider easily implementable variable step-size implicit-explicit （VSIMEX） linear multistep methods for time-dependent PDEs. Families of order-p, pstep VSIMEX schemes are constructed and analyzed, where p ranges from 1 to 4. The corresponding schemes are simple to implement and have the property that they reduce to the classical IMEX schemes whenever constant time step-sizes are imposed. The methods are validated on the Burgers＇ equation. These results demonstrate that by varying the time step-size, VSIMEX methods can outperform their fixed time step counterparts while still maintaining good numerical behavior.
基金The authors wish to thank the anonymous referees for their valuable comments and suggestions.The work is supported by the National Natural Science Foundation of China(Grant Nos.11671343,11701110)the Foundation for the Key Laboratory of Computational Physics,China(No.6142A05180103)the Scientific Research Fund of Science and Technology Department of Hunan Province in China(Grant No.2018WK4006).
文摘Implicit-explicit Runge-Kutta-Rosenbrock methods are proposed to solve nonlinear sti ordinary di erential equations by combining linearly implicit Rosenbrock methods with explicit Runge-Kutta methods.First,the general order conditions up to order 3 are obtained.Then,for the nonlinear sti initial-value problems satisfying the one-sided Lipschitz condition and a class of singularly perturbed initial-value problems,the corresponding errors of the implicit-explicit methods are analysed.At last,some numerical examples are given to verify the validity of the obtained theoretical results and the e ectiveness of the methods.
基金The authors would like to thank the referee and the editor for their valuable&constructive comments,which have greatly improved the article.This research was supported by the National Natural Science Foundation of China(Grant Nos.11871399,11471261,11101333,11302172,11571275)the Natural Science Foundation of Shaanxi(Grant No.2017JM 1005)the Fundamental Research Funds for the Central Universities of China(Grant Nos.31020180QD07&3102017zy041).
文摘This work develops a fully discrete implicit-explicit finite element scheme for a parabolicordinary system with a nonlinear reaction term which is known as the FitzHugh-Nagumo model from physiology.The first-order backward Euler discretization for the time derivative,and an implicit-explicit discretization for the nonlinear reaction term are employed for the model,with a simple linearization technique used to make the process of solving equations more efficient.The stability and convergence of the fully discrete implicit-explicit finite element method are proved,which shows that the FitzHugh-Nagumo model is accurately solved and the trajectory of potential transmission is obtained.The numerical results are also reported to verify the convergence results and the st ability of the proposed method.
基金supported by National Natural Science Foundation of China(Grant No.11471194)Department of Energy of USA(Grant No.DE-FG02-08ER25863)National Science Foundation of USA(Grant No.DMS-1418750)
文摘We consider the drift-diffusion （DD） model of one dimensional semiconductor devices, which is a system involving not only first derivative convection terms but also second derivative diffusion terms and a coupled Poisson potential equation. Optimal error estimates are obtained for both the semi-discrete and fully discrete local discontinuous Galerkin （LDG） schemes with smooth solutions. In the fully discrete scheme, we couple the implicit-explicit （IMEX） time discretization with the LDG spatial diseretization, in order to allow larger time steps and to save computational cost. The main technical difficulty in the analysis is to treat the inter-element jump terms which arise from the discontinuous nature of the numerical method and the nonlinearity and coupling of the models. A simulation is also performed to validate the analysis.
基金National Numerical Windtunnel Project NNW2019ZT4-B08, NSFC grant No. 12071455.
文摘In this paper,we present a mesh adaptation algorithm for the unsteady compressible Navier-Stokes equations under the framework of local discontinuous Galerkin methods coupled with implicit-explicit Runge-Kutta or spectral deferred correction time discretization methods.In both of the two high order semi-implicit time integration methods,the convective flux is treated explicitly and the viscous and heat fluxes are treated implicitly.The remarkable benefits of such semi-implicit temporal discretizations are that they can not only overcome the stringent time step restriction compared with time explicit methods,but also avoid the construction of the large Jacobian matrix as is done for fully implicit methods,thus are relatively easy to implement.To save computing time as well as capture the flow structures of interest accurately,a local mesh refinement(h-adaptive)technique,in which we present detailed criteria for selecting candidate elements and complete strategies to refine and coarsen them,is also applied for the Navier-Stokes equations.Numerical experiments are provided to illustrate the high order accuracy,efficiency and capabilities of the semi-implicit schemes in combination with adaptive local discontinuous Galerkin methods for the Navier-Stokes equations.
文摘Inspiral of binary black holes occurs over a time-scale of many orbits,far longer than the dynamical time-scale of the individual black holes.Explicit evolutions of a binary system therefore require excessively many time-steps to capture interesting dynamics.We present a strategy to overcome the Courant-Friedrichs-Lewy condition in such evolutions,one relying on modern implicit-explicit ODE solvers and multidomain spectral methods for elliptic equations.Our analysis considers the model problem of a forced scalar field propagating on a generic curved background.Nevertheless,we encounter and address a number of issues pertinent to the binary black hole problem in full general relativity.Specializing to the Schwarzschild geometry in KerrSchild coordinates,we document the results of several numerical experiments testing our strategy.
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.
