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Convergence analysis of SPH method on irregular particle distributions for the Poisson equation

Zhonghua Qiao, Yifan Wei

Abstract

The accuracy of particle approximation in Smoothed Particle Hydrodynamics (SPH) method decreases due to irregular particle distributions, especially for second-order derivatives. This study aims to enhance the accuracy of SPH method and analyze its convergence with irregular particle distributions. By establishing regularity conditions for particle distributions, we ensure that the local truncation error of traditional SPH formulations, including first and second derivatives, achieves second-order accuracy. Our proposed method, the volume reconstruction SPH method, guarantees these regularity conditions while preserving the discrete maximum principle. Benefiting from the discrete maximum principle, we conduct a rigorous global error analysis in the $L^\infty$-norm for the Poisson equation with variable coefficients, achieving second-order convergence. Numerical examples are presented to validate the theoretical findings.

Convergence analysis of SPH method on irregular particle distributions for the Poisson equation

Abstract

The accuracy of particle approximation in Smoothed Particle Hydrodynamics (SPH) method decreases due to irregular particle distributions, especially for second-order derivatives. This study aims to enhance the accuracy of SPH method and analyze its convergence with irregular particle distributions. By establishing regularity conditions for particle distributions, we ensure that the local truncation error of traditional SPH formulations, including first and second derivatives, achieves second-order accuracy. Our proposed method, the volume reconstruction SPH method, guarantees these regularity conditions while preserving the discrete maximum principle. Benefiting from the discrete maximum principle, we conduct a rigorous global error analysis in the -norm for the Poisson equation with variable coefficients, achieving second-order convergence. Numerical examples are presented to validate the theoretical findings.

Paper Structure

This paper contains 13 sections, 11 theorems, 103 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Traditional SPH and FPM
  3. Brief review of traditional SPH
  4. Approximating derivatives with FPM
  5. Regularity conditions for particle approximation
  6. Volume reconstruction for SPH
  7. Error estimates for the variable coefficient Poisson equation
  8. Numerical experiments
  9. Accuracy Test for interior particles
  10. Approximation accuracy of boundary particles
  11. CPU test
  12. Variable coefficient Poisson equation
  13. Conclusions

Key Result

Lemma 2.1

\newlabellem10 Here, $C$ is only dependent on the kernel function $\hat{W}(R)$ and the dimension parameter $d$.

Figures (7)

  • Figure 1: Randomly perturbed particles in 1D (top) 2D (bottom)
  • Figure 2: Gradient (left) and Laplace (right) approximation on uniform distribution in 2D
  • Figure 3: Truncation error of the gradient in 1D (left) and 2D (right)
  • Figure 4: Truncation error of the Laplace in 1D (left) and 2D (right)
  • Figure 5: Triangular element (left) and corresponding particle (right) distributions in 2D.
  • ...and 2 more figures

Theorems & Definitions (26)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 3.1
  • Proof 1
  • Theorem 3.2: Gradient approximation
  • Proof 2
  • Theorem 3.3: Laplace approximation
  • Proof 3
  • Theorem 3.4: Morris operator approximation
  • Proof 4
  • ...and 16 more