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A novel splitting method for Vlasov-Ampere

James A. Rossmanith, Christine Vaughan

TL;DR

The paper addresses the limited moment-order accuracy of traditional operator splitting for the 1D1V Vlasov–Ampère equation. It introduces a velocity-band moment-closure, where velocity space is partitioned into bands and evolved via local moments ${\mathbb M}_{\ell}^{(j)}$ for $\ell=0..4$, with a split into inter-band coupling (Problem A) and intra-band advection (Problem B) solved by Strang splitting. A Lax–Wendroff discontinuous Galerkin discretization with order ${M_O}$ advances the solution, achieving ${\mathcal{O}}(\Delta t^{M_O+1} + \Delta x^{M_O+1})$ accuracy for smooth solutions and preserving high-order accuracy of the global moments due to the splitting structure. Numerical tests on a manufactured solution, weak/strong Landau damping, and plasma sheath demonstrate fourth-order convergence and strong agreement with established results, indicating improved reliability for kinetic plasma simulations.

Abstract

Vlasov equations model the dynamics of plasma in the collisionless regime. A standard approach for numerically solving the Vlasov equation is to operator split the spatial and velocity derivative terms, allowing simpler time-stepping schemes to be applied to each piece separately (known as the Cheng-Knorr method). One disadvantage of such an operator split method is that the order of accuracy of fluid moments (e.g., mass, momentum, and energy) is restricted by the order of the operator splitting (second-order accuracy in the Cheng-Knorr case). In this work, we develop a novel approach that first represents the particle density function on a velocity mesh with a local fluid approximation in each discrete velocity band and then introduces an operator splitting that splits the inter-velocity band coupling terms from the dynamics within the discrete velocity band. The advantage is that the inter-velocity band coupling terms are only needed to achieve consistency of the full distribution functions, but the local fluid models within each band are sufficient to achieve high-order accuracy on global moments such as mass, momentum, and energy. The resulting scheme is verified on several standard Vlasov-Poisson test cases.

A novel splitting method for Vlasov-Ampere

TL;DR

The paper addresses the limited moment-order accuracy of traditional operator splitting for the 1D1V Vlasov–Ampère equation. It introduces a velocity-band moment-closure, where velocity space is partitioned into bands and evolved via local moments for , with a split into inter-band coupling (Problem A) and intra-band advection (Problem B) solved by Strang splitting. A Lax–Wendroff discontinuous Galerkin discretization with order advances the solution, achieving accuracy for smooth solutions and preserving high-order accuracy of the global moments due to the splitting structure. Numerical tests on a manufactured solution, weak/strong Landau damping, and plasma sheath demonstrate fourth-order convergence and strong agreement with established results, indicating improved reliability for kinetic plasma simulations.

Abstract

Vlasov equations model the dynamics of plasma in the collisionless regime. A standard approach for numerically solving the Vlasov equation is to operator split the spatial and velocity derivative terms, allowing simpler time-stepping schemes to be applied to each piece separately (known as the Cheng-Knorr method). One disadvantage of such an operator split method is that the order of accuracy of fluid moments (e.g., mass, momentum, and energy) is restricted by the order of the operator splitting (second-order accuracy in the Cheng-Knorr case). In this work, we develop a novel approach that first represents the particle density function on a velocity mesh with a local fluid approximation in each discrete velocity band and then introduces an operator splitting that splits the inter-velocity band coupling terms from the dynamics within the discrete velocity band. The advantage is that the inter-velocity band coupling terms are only needed to achieve consistency of the full distribution functions, but the local fluid models within each band are sufficient to achieve high-order accuracy on global moments such as mass, momentum, and energy. The resulting scheme is verified on several standard Vlasov-Poisson test cases.

Paper Structure

This paper contains 8 sections, 1 theorem, 20 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. A Novel Operator Splitting Method
  3. Numerical Method
  4. Numerical Examples
  5. Manufactured Solution
  6. Weak and Strong Landau Damping
  7. Plasma Sheath
  8. Conclusions

Key Result

proposition thmcounterproposition

Under the assumption that the distribution function, $f$, is zero at $v=V_{\text{min}}$ and $v=V_{\text{max}}$, the operator splitting technique shown in eqn:prob_A and eqn:prob_B clearly introduces an ${\mathcal{O}}\left(\Delta t^2\right)$ splitting error into the accuracy of the PDF, $f$, but ther

Figures (3)

  • Figure 1: Weak Landau damping problem. Panel (a) shows the exponential decay of the $L_2$-norm of the electric field and the line $0.06 e^{\gamma t}$ with the numerically computed decay rate $\gamma = -0.1536$, and Panel (b) shows the reconstructed PDF.
  • Figure 2: Strong Landau damping problem. Panel (a) shows the initial exponential decay of the $L_2$-norm of the electric field ($\gamma_1 = -0.2918$) followed by exponential growth due to nonlinear effects ($\gamma_2 = 0.08584$), and Panel (b) shows the reconstructed PDF.
  • Figure 3: Plasma sheath problem. Panel (a) shows the PDF, and Panel (b) shows the density.

Theorems & Definitions (2)

  • proposition thmcounterproposition
  • proof