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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> norm to enhance resolution while maintaining positivity and non-oscillatory properties.&#xa0;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.</p> Graphical abstract <p></p>

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

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  • 发表日期：

    2025年12月24日

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  • Takashi Minoshima,
  • Yosuke Matsumoto

摘要

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 \(L^2\) L 2 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.

Graphical abstract