In this paper,we present a quadratic auxiliary variable approach to develop a new class of energy-preserving Runge-Kutta methods for the Korteweg-de Vries equation.The quadratic auxiliary variable approach is first pr...In this paper,we present a quadratic auxiliary variable approach to develop a new class of energy-preserving Runge-Kutta methods for the Korteweg-de Vries equation.The quadratic auxiliary variable approach is first proposed to reformulate the original model into an equivalent system,which transforms the energy conservation law of the Korteweg-de Vries equation into two quadratic invariants of the reformulated system.Then the symplectic Runge-Kutta methods are directly employed for the reformulated model to arrive at a new kind of time semi-discrete schemes for the original problem.Under consistent initial conditions,the proposed methods are rigorously proved to maintain the original energy conservation law of the Korteweg-de Vries equation.In addition,the Fourier pseudo-spectral method is used for spatial discretization,resulting in fully discrete energy-preserving schemes.To implement the proposed methods effectively,we present a very efficient iterative technique,which not only greatly saves the calculation cost,but also achieves the purpose of practically preserving structure.Ample numerical results are addressed to confirm the expected order of accuracy,conservative property and efficiency of the proposed algorithms.展开更多
This paper presents a high order time discretization method by combining the semi-implicit spectral deferred correction method with energy stable linear schemes to simulate a series of phase field problems.We start wi...This paper presents a high order time discretization method by combining the semi-implicit spectral deferred correction method with energy stable linear schemes to simulate a series of phase field problems.We start with the linear scheme,which is based on the invariant energy quadratization approach and is proved to be linear unconditionally energy stable.The scheme also takes advantage of avoiding nonlinear iteration and the restriction of time step to guarantee the nonlinear system uniquely solvable.Moreover,the scheme leads to linear algebraic system to solve at each iteration,and we employ the multigrid solver to solve it efficiently.Numerical re-sults are given to illustrate that the combination of local discontinuous Galerkin(LDG)spatial discretization and the high order temporal scheme is a practical,accurate and efficient simulation tool when solving phase field problems.Namely,we can obtain high order accuracy in both time and space by solving some simple linear algebraic equations.展开更多
基金supported by the Foundation of Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems(Grant No.202002)the Natural Science Foundation of Jiangsu Province(Grant No.BK20180413)+4 种基金the National Natural Science Foundation of China(Grant Nos.11801269,12071216)supported by Science Challenge Project(Grant No.TZ2018002)National Science and TechnologyMajor Project(J2019-II-0007-0027)supported by the China Postdoctoral Science Foundation(Grant No.2020M670116)the Foundation of Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems(Grant No.202001).
文摘In this paper,we present a quadratic auxiliary variable approach to develop a new class of energy-preserving Runge-Kutta methods for the Korteweg-de Vries equation.The quadratic auxiliary variable approach is first proposed to reformulate the original model into an equivalent system,which transforms the energy conservation law of the Korteweg-de Vries equation into two quadratic invariants of the reformulated system.Then the symplectic Runge-Kutta methods are directly employed for the reformulated model to arrive at a new kind of time semi-discrete schemes for the original problem.Under consistent initial conditions,the proposed methods are rigorously proved to maintain the original energy conservation law of the Korteweg-de Vries equation.In addition,the Fourier pseudo-spectral method is used for spatial discretization,resulting in fully discrete energy-preserving schemes.To implement the proposed methods effectively,we present a very efficient iterative technique,which not only greatly saves the calculation cost,but also achieves the purpose of practically preserving structure.Ample numerical results are addressed to confirm the expected order of accuracy,conservative property and efficiency of the proposed algorithms.
基金Research of R.Guo is supported by NSFC grant No.11601490Research of Y.Xu is supported by NSFC grant No.11722112,91630207.
文摘This paper presents a high order time discretization method by combining the semi-implicit spectral deferred correction method with energy stable linear schemes to simulate a series of phase field problems.We start with the linear scheme,which is based on the invariant energy quadratization approach and is proved to be linear unconditionally energy stable.The scheme also takes advantage of avoiding nonlinear iteration and the restriction of time step to guarantee the nonlinear system uniquely solvable.Moreover,the scheme leads to linear algebraic system to solve at each iteration,and we employ the multigrid solver to solve it efficiently.Numerical re-sults are given to illustrate that the combination of local discontinuous Galerkin(LDG)spatial discretization and the high order temporal scheme is a practical,accurate and efficient simulation tool when solving phase field problems.Namely,we can obtain high order accuracy in both time and space by solving some simple linear algebraic equations.
