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变系数偏微分方程的Galerkin KPOD模型降阶方法

王丽1, 苗真2, 蒋耀林2   

  1. 1. 新疆大学数学与系统科学学院, 新疆 830046;
    2. 西安交通大学数学与统计学院, 陕西 710049
  • 收稿日期:2019-12-02 出版日期:2021-09-15 发布日期:2021-09-17
  • 基金资助:
    国家自然科学基金(61663043,11871393)和陕西省重点研发计划国际合作项目(2019KWZ-08).

王丽, 苗真, 蒋耀林. 变系数偏微分方程的Galerkin KPOD模型降阶方法[J]. 数值计算与计算机应用, 2021, 42(3): 226-236.

Wang Li, Miao Zhen, Jiang Yaolin. MODEL ORDER REDUCTION BASED ON GALERKIN KPOD FOR PARTIAL DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS[J]. Journal on Numerica Methods and Computer Applications, 2021, 42(3): 226-236.

MODEL ORDER REDUCTION BASED ON GALERKIN KPOD FOR PARTIAL DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS

Wang Li1, Miao Zhen2, Jiang Yaolin2   

  1. 1. Xinjiang University, College of Mathematics and System Science, Xinjiang 830046, China;
    2. Xi'an Jiaotong University, School of Mathematics Statistics, Shaanxi 710049, China
  • Received:2019-12-02 Online:2021-09-15 Published:2021-09-17
研究了变系数偏微分方程的Galerkin KPOD(Krylov Enhanced Proper Orthogonal Decomposition)模型降阶方法.首先基于Galerkin有限元理论建立变系数偏微分方程的空间离散格式,得到具有时变系数矩阵的常微分方程组,并对该常微分方程组应用KPOD模型降阶方法进行降阶并求解.其次,对降阶投影算子进行了分析,给出了Galerkin有限元解与GalerkinKPOD降阶解之间的误差界.最后用数值算例验证了变系数偏微分方程的Galerkin KPOD模型降阶求解方法的可行性,通过有限元离散解与GalerkinKPOD降阶解、GalerkinPOD降阶解之间的误差比较,体现GalerkinKPOD降阶方法的优势.
In this paper, model order reduction based on the Galerkin KPOD (Krylov Enhanced proper orthogonal decomposition) for partial differential equations with variable coefficient is studied. First, the spatial discrete scheme of partial differential equations with variable coefficient is established based on Galerkin finite element theory to obtain the ordinary differential system with time-varying coefficient matrix. Then the model order reduction method based on KPOD is applied to solve the ordinary differential system; Second, the reduced order projection operator is analyzed, and the error bound between Galerkin finite element solution and Galerkin KPOD reduced order solution is given. Finally, the feasibility of the Galerkin KPOD model order reduction method for partial differential equations with variable coefficient is verified by a numerical example. The advantage of Galerkin KPOD model order method is reflected by the error comparison with finite element solution between the Galerkin KPOD reduced order solution and the Galerkin POD reduced order solution.

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