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
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.
