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[02533] Reliable and Efficient Numerical Computation of Nonlocal Models

  • Session Date & Time :
    • 02533 (1/2) : 2C (Aug.22, 13:20-15:00)
    • 02533 (2/2) : 2D (Aug.22, 15:30-17:10)
  • Type : Proposal of Minisymposium
  • Abstract : Nonlocal models, which have proven effective in capturing long-range interactions in diverse applications, often involve integrals over a nonlocal horizon. Proper numerical discretization of these integrals is essential to ensure reliable and efficient simulations of the models. This requires addressing issues such as efficiently evaluating nonlocal integrals with domain-specific and computationally suitable meshes, as well as ensuring robustness as the size of the nonlocal horizon approaches zero. This minisymposium provides a platform for researchers to share their experiences and insights on designing reliable and efficient numerical schemes for nonlocal models.
  • Organizer(s) : Kuang Huang, Xiaobo Yin
  • Classification : 35R09, 65R20
  • Speakers Info :
    • Jiwei Zhang (Wuhan University)
    • Xiaobo Yin (Central China Normal University)
    • Xiaoping Zhang (Wuhan University)
    • Kuang Huang (Columbia University)
  • Talks in Minisymposium :
    • [03174] The Effect of Domain Truncation for Nonlocal Models and Asymptotically Compatible Schemes in Numerical Computation
      • Author(s) :
        • Xiaobo Yin (Central China Normal University)
        • Qiang Du (Columbia University)
        • Hehu Xie (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
        • Jiwei Zhang (Wuhan University)
      • Abstract : Many nonlocal models have adopted a finite and radially symmetric nonlocal interaction neighborhoods. When solving them numerically, it is sometimes convenient to adopt polygonal approximations of such interaction neighborhoods. A crucial question is, to what extent such approximations affect the nonlocal operators and the corresponding nonlocal solutions. While recent works have analyzed this issue for nonlocal operators in the case of a fixed horizon parameter, the question remains open in the case of a small or vanishing horizon parameter, which happens often in many practical applications and has significant impact on the reliability and robustness of nonlocal modeling and simulations. In this report, we are interested in addressing this issue and establishing the convergence of new nonlocal solutions by polygonal approximations to the local limit of the original nonlocal solutions. Our finding reveals that the new nonlocal solution does not converge to the correct local limit when the number of sides of polygons is uniformly bounded. On the other hand, if the number of sides tends to infinity, the desired convergence can be shown. We also apply this finding to discuss of the aysmptotically compatible property of the numerical schemes.
    • [03599] Asymptotical compatibility of a class of numerical schemes for a nonlocal traffic flow model
      • Author(s) :
        • Kuang Huang (Columbia University)
        • Qiang Du (Columbia University)
      • Abstract : This talk presents a study of numerical schemes for a nonlocal conservation law modeling traffic flows with nonlocal inter-vehicle interactions. We demonstrate the asymptotical compatibility of a class of finite volume schemes with suitable discretizations of the nonlocal integral. The numerical solutions produced by the schemes converge to the weak solution of the nonlocal model with a fixed nonlocal horizon and to the weak entropy solution of the respective local model as the mesh size and nonlocal horizon parameter go to zero simultaneously. Our findings provide insight into the development of robust numerical schemes for nonlocal conservation laws.
    • [03775] Global well-posedness of one new class of initial-boundary value problem on incompressible Navier-Stokes equations and the related models
      • Author(s) :
        • Shu Wang (Beijing University of Technology)
      • Abstract : The global well-posedness of the initial-boundary value problem on incompressible Navier-Stokes equations and the related models in the domain with the boundary is studied. The global existence of a class of weak solution to the initial boundary value problem to two/three-dimensional incompressible Navier-Stokes equation with the pressure-velocity relation at the boundary is obtained, and the global existence and uniqueness of the smooth solution to the corresponding problem in two-dimensional case is also established. Some extends to the corresponding incompressible fluid models such as Boussinesq equation/MHD equations and FSI models etc. are given.
    • [04201] High performance implementation of 3D FEM for nonlocal Poisson problem
      • Author(s) :
        • JIWEI ZHANG (Wuhan University )
      • Abstract : Nonlocality brings many challenges to the implementation of finite element methods (FEM) for nonlocal problems, such as large number of queries and invoke operations on the meshes. Besides, the interactions are usually limited to Euclidean balls, so direct numerical integrals often introduce numerical errors. The issues of interactions between the ball and finite elements have to be carefully dealt with, such as using ball approximation strategies. In this talk, an efficient representation and construction methods for approximate balls are presented based on combinatorial map, and an efficient parallel algorithm is also designed for assembly of nonlocal linear systems. Specifically, a new ball approximation method based on Monte Carlo integrals, i.e., the fullcaps method, is also proposed to compute numerical integrals over the intersection region of an element with the ball.
    • [04684] Asymptotically compatible collocation scheme for nonlocal diffusion problems
      • Author(s) :
        • Xiaoping Zhang (Wuhan University)
      • Abstract : In this talk, I will introduce a collocation scheme for solving nonlocal diffusion problems with general kernels, including the steady and time-dependent state. The convergence analysis is rigorously conducted for both cases. Besides, The Asymptotical compatibility is proved for the steady case. Numerical experiments are presented to validate the theoretical results.