In this paper,we develop a general framework for constructing higher-order,unconditionally energydecreasing exponential time differencing Runge-Kutta(ETDRK)methods applicable to a range of gradient flows.Specifically,...In this paper,we develop a general framework for constructing higher-order,unconditionally energydecreasing exponential time differencing Runge-Kutta(ETDRK)methods applicable to a range of gradient flows.Specifically,we identify conditions sufficient for ETDRK schemes to maintain the original energy dissipation.Our analysis reveals that the widely-employed third-and fourth-order ETDRK schemes fail to meet these conditions.To address this,we introduce new third-order ETDRK schemes,designed with appropriate stabilization,which satisfy these conditions and thus guarantee the unconditional energy decay property.We conduct extensive numerical experiments with these new schemes to verify their accuracy,stability,behavior under large time steps,long-term evolution,and adaptive time-stepping strategy across various gradient flows.This study offers the first framework to examine the unconditional energy stability of high-order ETDRK methods,and we are optimistic that our framework will enable the development of ETDRK schemes beyond the third order that are unconditionally energy stable.展开更多
We present the new predictor-corrector methods for systems of nonlinear differential equations, based on the method of exponential time differencing. We compare the present schemes with the explicit multistep exponent...We present the new predictor-corrector methods for systems of nonlinear differential equations, based on the method of exponential time differencing. We compare the present schemes with the explicit multistep exponential time differencing and Adams–Bashforth–Moulton method. The numerical results show that the schemes are more accurate and more efficient than Adams predictor-corrector method. The exponential time differencing method has been developed and perfected by the present studies.展开更多
The Filon-type quadrature is efficient for highly oscillatory functions - Fourier transforms. Based on Cox and Matthews' ETD schemes, the higher order single step exponential time differencing schemes are presente...The Filon-type quadrature is efficient for highly oscillatory functions - Fourier transforms. Based on Cox and Matthews' ETD schemes, the higher order single step exponential time differencing schemes are presented based on the Filon-type integration and the A-stability of the two-order Adams-Bashforth exponential time differencing scheme is considered. The effectiveness and accuracy of the schemes is tested.展开更多
Reaction-diffusion equations modeling Predator-Prey interaction are of current interest. Standard approaches such as first-order (in time) finite difference schemes for approximating the solution are widely spread. Th...Reaction-diffusion equations modeling Predator-Prey interaction are of current interest. Standard approaches such as first-order (in time) finite difference schemes for approximating the solution are widely spread. Though, this paper shows that recent advance methods can be more favored. In this work, we have incorporated, throughout numerical comparison experiments, spectral methods, for the space discretization, in conjunction with second and fourth-order time integrating methods for approximating the solution of the reaction-diffusion differential equations. The results have revealed that these methods have advantages over the conventional methods, some of which to mention are: the ease of implementation, accuracy and CPU time.展开更多
In this paper,we prove the optimal error estimates in L2 norm of the semidiscrete local discontinuous Galerkin(LDG)method for the thin film epitaxy problem without slope selection.To relax the severe time step restric...In this paper,we prove the optimal error estimates in L2 norm of the semidiscrete local discontinuous Galerkin(LDG)method for the thin film epitaxy problem without slope selection.To relax the severe time step restriction of explicit time marching methods,we employ a class of exponential time differencing(ETD)schemes for time integration,which is based on a linear convex splitting principle.Numerical experiments of the accuracy and long time simulations are given to show the efficiency and capability of the proposed numerical schemes.展开更多
In a previous paper,a technique was suggested to avoid order reduction with any explicit exponential Runge-Kutta method when integrating initial boundary value nonlinear problems with time-dependent boundary condition...In a previous paper,a technique was suggested to avoid order reduction with any explicit exponential Runge-Kutta method when integrating initial boundary value nonlinear problems with time-dependent boundary conditions.In this paper,we significantly simplify the full discretization formulas to be applied under conditions which are nearly always satisfied in practice.Not only a simpler linear combination of'j-functions is given for both the stages and the solution,but also the information required on the boundary is so much simplified that,in order to get local order three,it is no longer necessary to resort to numerical differentiation in space.In many cases,even to get local order 4.The technique is then shown to be computationally competitive against other widely used methods with high enough stiff order through the standard method of lines.展开更多
In the paper,we propose a novel linearly implicit structure-preserving algorithm,which is derived by combing the invariant energy quadratization approach with the exponential time differencing method,to construct effi...In the paper,we propose a novel linearly implicit structure-preserving algorithm,which is derived by combing the invariant energy quadratization approach with the exponential time differencing method,to construct efficient and accurate time discretization scheme for a large class of Hamiltonian partial differential equations(PDEs).The proposed scheme is a linear system,and can be solved more efficient than the original energy-preserving ex-ponential integrator scheme which usually needs nonlinear iterations.Various experiments are performed to verify the conservation,efficiency and good performance at relatively large time step in long time computations.展开更多
基金supported by National Natural Science Foundation of China(Grant No.12371409)supported by National Natural Science Foundation of China(Grant No.12271240)+1 种基金National Natural Science Foundation of China/Hong Kong Research Grants Council Joint Research Scheme(Grant No.11961160718)the Shenzhen Natural Science Fund(Grant No.RCJC20210609103819018)。
文摘In this paper,we develop a general framework for constructing higher-order,unconditionally energydecreasing exponential time differencing Runge-Kutta(ETDRK)methods applicable to a range of gradient flows.Specifically,we identify conditions sufficient for ETDRK schemes to maintain the original energy dissipation.Our analysis reveals that the widely-employed third-and fourth-order ETDRK schemes fail to meet these conditions.To address this,we introduce new third-order ETDRK schemes,designed with appropriate stabilization,which satisfy these conditions and thus guarantee the unconditional energy decay property.We conduct extensive numerical experiments with these new schemes to verify their accuracy,stability,behavior under large time steps,long-term evolution,and adaptive time-stepping strategy across various gradient flows.This study offers the first framework to examine the unconditional energy stability of high-order ETDRK methods,and we are optimistic that our framework will enable the development of ETDRK schemes beyond the third order that are unconditionally energy stable.
基金The project supported by National Natural Science Foundation of China under Grant No.19902002
文摘We present the new predictor-corrector methods for systems of nonlinear differential equations, based on the method of exponential time differencing. We compare the present schemes with the explicit multistep exponential time differencing and Adams–Bashforth–Moulton method. The numerical results show that the schemes are more accurate and more efficient than Adams predictor-corrector method. The exponential time differencing method has been developed and perfected by the present studies.
基金Projects(02JJY2006, 03JJY2001) supported by Natural Science Foundation of Hunan Province project supported by JSPS Fellowship Research Program
文摘The Filon-type quadrature is efficient for highly oscillatory functions - Fourier transforms. Based on Cox and Matthews' ETD schemes, the higher order single step exponential time differencing schemes are presented based on the Filon-type integration and the A-stability of the two-order Adams-Bashforth exponential time differencing scheme is considered. The effectiveness and accuracy of the schemes is tested.
文摘Reaction-diffusion equations modeling Predator-Prey interaction are of current interest. Standard approaches such as first-order (in time) finite difference schemes for approximating the solution are widely spread. Though, this paper shows that recent advance methods can be more favored. In this work, we have incorporated, throughout numerical comparison experiments, spectral methods, for the space discretization, in conjunction with second and fourth-order time integrating methods for approximating the solution of the reaction-diffusion differential equations. The results have revealed that these methods have advantages over the conventional methods, some of which to mention are: the ease of implementation, accuracy and CPU time.
基金This work is supported by NSFC grants No.11601490.
文摘In this paper,we prove the optimal error estimates in L2 norm of the semidiscrete local discontinuous Galerkin(LDG)method for the thin film epitaxy problem without slope selection.To relax the severe time step restriction of explicit time marching methods,we employ a class of exponential time differencing(ETD)schemes for time integration,which is based on a linear convex splitting principle.Numerical experiments of the accuracy and long time simulations are given to show the efficiency and capability of the proposed numerical schemes.
基金supported by Junta de Castilla y León and Feder(Project VA169P20).
文摘In a previous paper,a technique was suggested to avoid order reduction with any explicit exponential Runge-Kutta method when integrating initial boundary value nonlinear problems with time-dependent boundary conditions.In this paper,we significantly simplify the full discretization formulas to be applied under conditions which are nearly always satisfied in practice.Not only a simpler linear combination of'j-functions is given for both the stages and the solution,but also the information required on the boundary is so much simplified that,in order to get local order three,it is no longer necessary to resort to numerical differentiation in space.In many cases,even to get local order 4.The technique is then shown to be computationally competitive against other widely used methods with high enough stiff order through the standard method of lines.
基金supported by the National Natural Science Foundation of China(Grant Nos.12171245,11971416,11971242,12301508)by the Natural Science Foundation of Henan Province(Grant No.222300420280)+1 种基金by the Natural Science Foundation of Hunan Province(Grant No.2023JJ40656)by the Scientific Research Fund of Xuchang University(Grant No.2024ZD010).
文摘In the paper,we propose a novel linearly implicit structure-preserving algorithm,which is derived by combing the invariant energy quadratization approach with the exponential time differencing method,to construct efficient and accurate time discretization scheme for a large class of Hamiltonian partial differential equations(PDEs).The proposed scheme is a linear system,and can be solved more efficient than the original energy-preserving ex-ponential integrator scheme which usually needs nonlinear iterations.Various experiments are performed to verify the conservation,efficiency and good performance at relatively large time step in long time computations.
