A high-order weighted positive and flux conservative method for the Vlasov equation

TL;DR

<3-5 sentence high-level summary> This paper addresses the challenge of solving the Vlasov equation with high fidelity while preserving physical constraints. It introduces the WPFC method, a fifth-order, positivity-preserving, non-oscillatory, conservative semi-Lagrangian scheme built on the PFC framework, employing L2-norm-based nonlinear weights to boost resolution. An approximate dispersion analysis demonstrates favorable spectral properties, and numerical experiments on linear advection and Vlasov-Ampère/Maxwell problems show improved entropy conservation and reduced numerical diffusion. The approach offers a practical high-order alternative for kinetic plasma simulations, with a trade-off in computational cost.

Abstract

We present a high-order conservative, positivity-preserving, and non-oscillatory scheme for solving the Vlasov equation. The scheme attains formal fifth-order accuracy through a convex combination of positive and non-oscillatory polynomials in substencils. Nonlinear weights for these polynomials are formulated that assign higher priority to substencils with larger L2 norm to enhance resolution while maintaining positivity and non-oscillatory properties. An approximate dispersion relation indicates that the spectral properties of the present scheme outperform those of an underlying fifth-order scheme and even surpass those of a seventh-order scheme in certain wavenumber ranges. We apply this scheme to the one-dimensional Vlasov-Ampere equations and the two-dimensional Vlasov-Maxwell equations, and demonstrate high-resolution simulations with improved conservation of entropy.

Figures (9)

• Figure 1: Approximate dispersion relation. (a) Real and (b) imaginary parts of the modified wavenumber in the present scheme are shown in red lines. For comparison, results from linear first-, fifth-, and seventh-order schemes are shown in black, green, and blue lines. Dash-dotted lines indicate the exact solution.
• Figure 2: Difference in (a) the dispersion error and (b) the dissipation error between the present scheme and the linear seventh-order scheme. The spectral properties of the present scheme are superior when the difference is negative in (a) and positive in (b), respectively.
• Figure 3: Linear advection over 10 periods obtained from fifth-order conservative semi-Lagrangian schemes: (a) the linear scheme, (b) the monotonicity-preserving scheme, and (c) the present scheme. Solid lines indicate the exact solution.
• Figure 4: Numerical experiment of the two stream instability. (a,b) Electron phase space distribution at $t=30 \omega_{\rm pe}^{-1}$ and $800 \omega_{\rm pe}^{-1}$. (c,d) Time profiles of the total energy and entropy normalized by their initial values, obtained from the third-order PFC scheme (black lines), the fifth- and seventh-order monotonicity-preserving schemes (green and purple lines), and the present scheme (red lines), respectively.
• Figure 5: Numerical experiment of the bump-on-tail instability. The figure follows the same format as Figure \ref{['fig:twost']}.