Banach空间中复合刚性Volterra泛函微分方程隐显Euler方法的稳定性分析  

Stability Analysis of Implicit-explicit Euler Method for Composite Stiff Volterra Functional Differential Equations in Banach Space

作　　者：龙滔 余越昕 LONG Tao;YU Yuexin(School of Mathematics and Computational Science,Xiangtan University,Xiangtan 411105)

机构地区：[1]湘潭大学数学与计算科学学院,湘潭411105

出　　处：《工程数学学报》2023年第6期929-940,共12页Chinese Journal of Engineering Mathematics

基　　金：国家自然科学基金(12271367);湖南省教育厅重点项目(21A0115).

摘　　要：刚性泛函微分方程数值方法的研究大多是在内积空间中基于单边Lipschitz常数具有适度大小的条件下进行;然而对于某些刚性问题,其单边Lipschitz常数却不可避免地取非常巨大的正值。因此有必要突破内积空间和单边Lipschitz常数的限制,直接在Banach空间中探讨相应的数值方法。针对Banach空间中的非线性复合刚性Volterra泛函微分方程,对其非刚性部分采用显式Euler方法求解,刚性部分采用隐式Euler方法求解,得到了求解该问题的隐显Euler方法,论证了方法的稳定性和渐近稳定性。数值试验结果验证了所获理论的正确性。The study of numerical methods for stiff functional differential equations is mostly carried out in the inner product space based on the assumption that the one-sided Lipschitz constant has a moderate size,whereas for some stiff problems,the one-sided Lipschitz constant inevitably takes very large positive values.Therefore,it is necessary to break through the limitations of inner product space and one-sided Lipschitz constant and study the related numerical methods directly in the Banach space.For the nonlinear composite stiff Volterra functional differential equation in Banach space,the non-stiff part is solved by the explicit Euler method and the stiff part is solved by the implicit Euler method,obtained from which is the implicit-explicit Euler method for solving the problem.The stability and asymptotic stability about the proposed method are established,and the numerical results verify the correctness of the obtained theory.

关 键 词：复合刚性微分方程 稳定性 渐近稳定性 隐显Euler方法 BANACH空间 

分 类 号：O241.81[理学—计算数学]