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VIM-based dynamic sparse grid approach to partial differential equations.

Shu-Li Mei1

  • 1College of Information and Electrical Engineering, China Agricultural University, P.O. Box 53, East Campus, 17 Qinghua Donglu Road, Haidian District, Beijing 100083, China.

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A new dynamic sparse grid method combines variational iteration method (VIM) with sparse grid theory for nonlinear partial differential equations (PDEs). This approach offers improved accuracy, particularly for problems with Nuemann boundary conditions.

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Area of Science:

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

Background:

  • Nonlinear partial differential equations (PDEs) are fundamental in modeling complex phenomena.
  • Traditional numerical methods can face challenges with accuracy and efficiency for nonlinear PDEs.
  • Existing methods like interval wavelet collocation may have limitations, especially with specific boundary conditions.

Purpose of the Study:

  • To introduce a novel dynamic sparse grid approach for solving nonlinear PDEs.
  • To enhance the accuracy and efficiency of numerical solutions for nonlinear PDEs.
  • To address limitations of existing methods, particularly for Nuemann boundary conditions.

Main Methods:

  • The proposed method integrates the variational iteration method (VIM) with sparse grid theory.
  • A multilevel interpolation operator is constructed using sparse grid principles.
  • The VIM is adapted using the precise integration method (PIM) to solve discretized ODEs.
  • A dynamic selection scheme for inner and outer grid points is implemented.

Main Results:

  • The dynamic sparse grid approach demonstrates superior performance compared to traditional wavelet collocation methods.
  • The method shows particular effectiveness in solving nonlinear PDEs with Nuemann boundary conditions.
  • Numerical experiments validate the accuracy and efficiency of the proposed technique.

Conclusions:

  • The combined VIM and dynamic sparse grid method provides an effective and accurate solution for nonlinear PDEs.
  • This approach offers a significant improvement over existing methods, especially for challenging boundary conditions.
  • The dynamic grid point selection enhances the method's adaptability and performance.