一致二阶精度的自启动单解时域积分器  

THE SELF-STARTING SINGLE-SOLVE TIME INTEGRATORS WITH IDENTICAL SECOND-ORDER ACCURACY

作　　者：李金泽 刘耀坤 崔乃刚[1] 于开平[1] Li Jinze;Liu Yaokun;Cui Naigang;Yu Kaiping(School of Astronautics,Harbin Institute of Technology,Harbin 150001,China)

机构地区：[1]哈尔滨工业大学航天学院,哈尔滨150001

出　　处：《力学学报》2025年第3期767-781,共15页Chinese Journal of Theoretical and Applied Mechanics

基　　金：国家资助博士后研究人员计划(GZC20233464);中国博士后科学基金(2024M764165);黑龙江省博士后基金(LBH-Z23153);国家自然科学基金(12272105)资助项目.

摘　　要：时域积分器是分析大型一阶动力系统常用的数值求解技术且具有竞争性的时域积分器应实现自启动、单解、一致二阶精度和可控数值阻尼.在无辅助技术的加持下,现有的自启动单解时域积分器未能同时实现可控数值阻尼和一致二阶精度.为了突破该限制,首先构造了一类自启动单解时域积分器,然后推导了时域积分器实现一致二阶精度的代数条件,最后提出了一致二阶精度的自启动单解隐式和显式时域积分器.新隐式积分器是无条件稳定的且将数值放大矩阵在高频极限下的特征值作为两个用户指定参数进而灵活地控制数值高频阻尼;当两个用户指定参数相等时,两参数的隐式积分器退化为最优数值阻尼和最小误差常数的单参数积分器.新显式积分器是条件稳定的且将分岔点处的谱半径值作为用户指定参数进而控制数值阻尼;使用推荐的参数值时,显式积分器可以在条件稳定域和求解精度上达到合理平衡.性能比较和数值算例都验证了理论分析并凸显了本文自启动单解时域积分器在精度和耗散能力上的优势.Time integrators are essential numerical methods frequently employed for analyzing large-scale first-order dynamical systems.To ensure the competitiveness of a time integrator,it must meet several key requirements,including being self-starting,single-solve,possessing identical second-order accuracy,and offering controllable numerical dampings.However,current self-starting single-solve time integrators that do not employ auxiliary techniques face a significant limitation:they are unable to simultaneously achieve both controllable numerical dampings and identical second-order accuracy.To address this shortcoming,this paper takes several critical steps.First,a new family of selfstarting single-solve time integrators is introduced.These integrators are designed with the goal of achieving both the required numerical dampings and second-order accuracy.Following this,algebraic conditions are derived to ensure that the time integrators meet the criterion of identical second-order accuracy.In the final part of the study,self-starting single-solve implicit and explicit time integrators are proposed,both of which fulfill the condition of identical second-order accuracy.The newly developed implicit integrators are particularly notable for their unconditional stability.One of the distinguishing features of these implicit integrators is their ability to control numerical high-frequency dampings,which is achieved through two user-defined parameters.These parameters influence the eigenvalues of the numerical amplification matrix in the high-frequency limit,allowing for greater flexibility in damping control.When these two parameters are set equal,the implicit integrator simplifies into a single-parameter scheme that strikes an optimal balance between numerical dampings and minimizes the associated error constant.The new explicit integrators,on the other hand,are conditionally stable and incorporate a user-defined spectral radius at the bifurcation point,which serves to regulate the numerical dampings.By choosing the appropriate

关 键 词：一阶动力系统 一致二阶精度 单解 时域积分器 数值阻尼 

分 类 号：O302[理学—力学] O241.81