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

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 $$L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> 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