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Explicit relaxation Particle-in-Cell methods for Vlasov-Poisson equations with a strong magnetic field

Lina Wang, Bin Wang

Abstract

In this work, we present a novel family of explicit relaxation Particle-in-Cell (ER-PIC) methods for the Vlasov-Poisson equation with a strong magnetic field. These schemes achieve exact energy conservation by combining a splitting framework with the dynamic updating of a relaxation parameter at each time step. Using an averaging technique, we rigorously establish second-order error bounds for the Strang-type ER-PIC method and uniform first order accuracy in position for the Lie-Trotter ER-PIC scheme. Numerical experiments across the fluid, finite Larmor radius, and diffusion regimes confirm the accuracy and energy conservation of our methods.

Explicit relaxation Particle-in-Cell methods for Vlasov-Poisson equations with a strong magnetic field

Abstract

In this work, we present a novel family of explicit relaxation Particle-in-Cell (ER-PIC) methods for the Vlasov-Poisson equation with a strong magnetic field. These schemes achieve exact energy conservation by combining a splitting framework with the dynamic updating of a relaxation parameter at each time step. Using an averaging technique, we rigorously establish second-order error bounds for the Strang-type ER-PIC method and uniform first order accuracy in position for the Lie-Trotter ER-PIC scheme. Numerical experiments across the fluid, finite Larmor radius, and diffusion regimes confirm the accuracy and energy conservation of our methods.

Paper Structure

This paper contains 14 sections, 6 theorems, 133 equations, 13 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Explicit relaxation Particle-in-Cell (ER-PIC) method
  3. PIC framework
  4. Construction of ER-PIC method
  5. Global convergence
  6. Main result
  7. Transformation of RS2-PIC scheme
  8. A rough error estimate
  9. Optimal error estimate
  10. Applications to Vlasov-Poisson system under various scalings
  11. Vlasov-Poisson equation in fluid scaling
  12. Vlasov-Poisson equation in Larmor scaling
  13. Vlasov-Poisson equation in Diffusion scaling
  14. Conclusion

Key Result

Theorem 2.1

For a sufficiently small time step size $h>0$, the relaxation parameter $\gamma_{n}$ in equ-10-1-6 is second-order accurate. Consequently, the RSV method is also second-order.

Figures (13)

  • Figure 1: Example 1. Contour plot of quantity $\rho(t,\mathbf{x})-n_{i}$ at different $t$ with $\varepsilon=0.005$ for RS2-PIC.
  • Figure 2: Example 1. Contour plot of quantity $\chi(t,\mathbf{v})$ at different $t$ with $\varepsilon=0.005$ for RS2-PIC.
  • Figure 3: Example 1. Errors of RS1-PIC (left) and RS2-PIC (right) with respect to $\varepsilon$ under different $h$ about $\rho$ and $\rho_{\mathbf{v}}$.
  • Figure 4: Example 1. Energy error of RS1-PIC (left) and RS2-PIC (right) with step size $h=0.1$ until $T=100$.
  • Figure 5: Example 1. Evolution of relaxation parameter $\gamma_{n}$ by using RS2-PIC with $h=0.02$ .
  • ...and 8 more figures

Theorems & Definitions (11)

  • Theorem 2.1
  • Proposition 2.2
  • Proof 1
  • Theorem 3.1
  • Lemma 3.2
  • Proof 2
  • Proposition 3.3
  • Proof 3
  • Theorem 3.4
  • Proof 4
  • ...and 1 more