Thermal phase change problems are widespread in mathematics,nature,and science.They are particularly useful in simulating the phenomena of melting and solidification in materials science.In this paper we propose a nov...Thermal phase change problems are widespread in mathematics,nature,and science.They are particularly useful in simulating the phenomena of melting and solidification in materials science.In this paper we propose a novel class of arbitrarily high-order and unconditionally energy stable schemes for a thermal phase changemodel,which is the coupling of a heat transfer equation and a phase field equation.The unconditional energy stability and consistency error estimates are rigorously proved for the proposed schemes.A detailed implementation demonstrates that the proposed method requires only the solution of a system of linear elliptic equations at each time step,with an efficient scheme of sufficient accuracy to calculate the solution at the first step.It is observed from the comparison with the classical explicit Runge-Kutta method that the new schemes allow to use larger time steps.Adaptive time step size strategies can be applied to further benefit from this unconditional stability.Numerical experiments are presented to verify the theoretical claims and to illustrate the accuracy and effectiveness of our method.展开更多
In this paper,we present a quadratic auxiliary variable(QAV)technique to develop a novel class of arbitrarily high-order energy-preserving algorithms for the Camassa-Holm equation.The QAV approach is first utilized to...In this paper,we present a quadratic auxiliary variable(QAV)technique to develop a novel class of arbitrarily high-order energy-preserving algorithms for the Camassa-Holm equation.The QAV approach is first utilized to transform the original equation into a reformulated QAV system with a consistent initial condition.Then the reformulated QAV system is discretized by applying the Fourier pseudo-spectral method in space and the symplectic Runge-Kutta methods in time,which arrives at a class of fully discrete schemes.Under the consistent initial condition,they can be rewritten as a new fully discrete system by eliminating the introduced auxiliary variable,which is rigorously proved to be energy-preserving and symmetric.Ample numerical experiments are conducted to confirm the expected order of accuracy,conservative property and efficiency of the proposed methods.The presented numerical strategy makes it possible to directly apply a special class of Runge-Kutta methods to develop energy-preserving algorithms for a general conservative system with any polynomial energy.展开更多
The Sasa-satsuma(SS)dynamical equation interpret propagation of ultra-short and femto-second pulses in optical fibers.This dynamical model has important physical significance.In this article,two mathematical technique...The Sasa-satsuma(SS)dynamical equation interpret propagation of ultra-short and femto-second pulses in optical fibers.This dynamical model has important physical significance.In this article,two mathematical techniques namely,improved F-expansion and improved aux-iliary methods are utilized to construct the several types of solitons such as dark soliton,bright soliton,periodic soliton,Elliptic function and solitary waves solutions of Sasa-satsuma dynamical equation.These results have imperative applications in sciences and other fields,and construc-tive to recognize the physical structure of this complex dynamical model.The computing work and obtained results show the infuence and effectiveness of current methods.展开更多
文摘Thermal phase change problems are widespread in mathematics,nature,and science.They are particularly useful in simulating the phenomena of melting and solidification in materials science.In this paper we propose a novel class of arbitrarily high-order and unconditionally energy stable schemes for a thermal phase changemodel,which is the coupling of a heat transfer equation and a phase field equation.The unconditional energy stability and consistency error estimates are rigorously proved for the proposed schemes.A detailed implementation demonstrates that the proposed method requires only the solution of a system of linear elliptic equations at each time step,with an efficient scheme of sufficient accuracy to calculate the solution at the first step.It is observed from the comparison with the classical explicit Runge-Kutta method that the new schemes allow to use larger time steps.Adaptive time step size strategies can be applied to further benefit from this unconditional stability.Numerical experiments are presented to verify the theoretical claims and to illustrate the accuracy and effectiveness of our method.
基金Yuezheng Gong’s work is partially supported by the Foundation of Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems(Grant No.202002)the Fundamental Research Funds for the Central Universities(Grant No.NS2022070)+7 种基金the Natural Science Foundation of Jiangsu Province(Grant No.BK20220131)the National Natural Science Foundation of China(Grants Nos.12271252 and 12071216)Qi Hong’s work is partially supported by the National Natural Science Foundation of China(Grants No.12201297)the Foundation of Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems(Grant No.202001)Chunwu Wang’s work is partially supported by Science Challenge Project(Grant No.TZ2018002)National Science and Technology Major Project(J2019-II-0007-0027)Yushun Wang’s work is partially supported by the National Key Research and Development Program of China(Grant No.2018YFC1504205)the National Natural Science Foundation of China(Grants No.12171245).
文摘In this paper,we present a quadratic auxiliary variable(QAV)technique to develop a novel class of arbitrarily high-order energy-preserving algorithms for the Camassa-Holm equation.The QAV approach is first utilized to transform the original equation into a reformulated QAV system with a consistent initial condition.Then the reformulated QAV system is discretized by applying the Fourier pseudo-spectral method in space and the symplectic Runge-Kutta methods in time,which arrives at a class of fully discrete schemes.Under the consistent initial condition,they can be rewritten as a new fully discrete system by eliminating the introduced auxiliary variable,which is rigorously proved to be energy-preserving and symmetric.Ample numerical experiments are conducted to confirm the expected order of accuracy,conservative property and efficiency of the proposed methods.The presented numerical strategy makes it possible to directly apply a special class of Runge-Kutta methods to develop energy-preserving algorithms for a general conservative system with any polynomial energy.
文摘The Sasa-satsuma(SS)dynamical equation interpret propagation of ultra-short and femto-second pulses in optical fibers.This dynamical model has important physical significance.In this article,two mathematical techniques namely,improved F-expansion and improved aux-iliary methods are utilized to construct the several types of solitons such as dark soliton,bright soliton,periodic soliton,Elliptic function and solitary waves solutions of Sasa-satsuma dynamical equation.These results have imperative applications in sciences and other fields,and construc-tive to recognize the physical structure of this complex dynamical model.The computing work and obtained results show the infuence and effectiveness of current methods.
