A High-Order Semi-Lagrangian Finite Difference Method for Nonlinear Vlasov and BGK Models

In this paper, we propose a new conservative high-order semi-Lagrangian finite difference (SLFD) method to solve linear advection equation and the nonlinear Vlasov and BGK models. The finite difference scheme has better computational flexibility by working with point values, especially when working with high-dimensional problems in an operator splitting setting. The reconstruction procedure in the proposed SLFD scheme is motivated from the SL finite volume scheme. In particular, we define a new sliding average function, whose cell averages agree with point values of the underlying function. By developing the SL finite volume scheme for the sliding average function, we derive the proposed SLFD scheme, which is high-order accurate, mass conservative and unconditionally stable for linear problems. The performance of the scheme is showcased by linear transport applications, as well as the nonlinear Vlasov-Poisson and BGK models. Furthermore, we apply the Fourier stability analysis to a fully discrete SLFD scheme coupled with diagonally implicit Runge-Kutta (DIRK) method when applied to a stiff two-velocity hyperbolic relaxation system. Numerical stability and asymptotic accuracy properties of DIRK methods are discussed in theoretical and computational aspects.