Conservative Semi-Lagrangian Finite Difference WENO Formulations with Applications to the Vlasov Equation

Abstract

In this paper, we propose a new conservative semi-Lagrangian (SL) finite difference (FD) WENO scheme for linear advection equations, which can serve as a base scheme for the Vlasov equation by Strang splitting [4]. The reconstruction procedure in the proposed SL FD scheme is the same as the one used in the SL finite volume (FV) WENO scheme [3]. However, instead of inputting cell averages and approximate the integral form of the equation in a FV scheme, we input point values and approximate the differential form of equation in a FD spirit, yet retaining very high order (fifth order in our experiment) spatial accuracy. The advantage of using point values, rather than cell averages, is to avoid the second order spatial error, due to the shearing in velocity (v) and electrical field (E) over a cell when performing the Strang splitting to the Vlasov equation. As a result, the proposed scheme has very high spatial accuracy, compared with second order spatial accuracy for Strang split SL FV scheme for solving the Vlasov-Poisson (VP) system. We perform numerical experiments on linear advection, rigid body rotation problem; and on the Landau damping and two-stream instabilities by solving the VP system. For comparison, we also apply (1) the conservative SL FD WENO scheme, proposed in [22] for incompressible advection problem, (2) the conservative SL FD WENO scheme proposed in [21] and (3) the non-conservative version of the SL FD WENO scheme in [3] to the same test problems. The performances of different schemes are compared by the error table, solution resolution of sharp interface, and by tracking the conservation of physical norms, energies and entropies, which should be physically preserved.

An h-Adaptive RKDG Method for the Vlasov–Poisson System

“…However, it suffers statistical noise O (1/ √ N ) with N being the number of particles. There are highly accurate mesh-based semi-Lagrangian [5,11,17,[24][25][26]28,32,33] and Eulerian [1,8,15,21,23,34,35,37] methods, which have been shown to be advantageous due to their efficiency and effectiveness in resolving rich solution structures. The semi-Lagrangian method is designed by propagating information along characteristic curves.…”

Conservative and Easily Implemented Finite Volume Semi-Lagrangian WENO Methods for 1D and 2D Hyperbolic Conservation Laws

“…Actually, there are two reconstruction procedures needed to obtain the numerical flux at cell edge 1 2 i x + . It is noted that, for linear equation with constant coefficient, the formulas of average flux (17) and (21) are totally same as ones in [16]. However, these two methods are devised by totally different procedure.…”

“…For the nonlinear scalar equation, we first use Rankine-Hugoniot jump condition to identify the upwinding direction, then compute the average flux by (17) or (21). In addition, an entropy fix is used to modify the average flux when rarefaction wave appears.…”

“…Recently, in the papers [14] [15] [16] [17], the authors combined the idea of semi-Lagrangian with WENO reconstruction for solving advection problems. The semi-Lagrangian methods above depend on the characteristic tracing.…”

The semi-Lagrangian method on curvilinear grids

“…There have also been works on positivity preserving conservative Discontinuous Galerkin methods [15,16] and also on semi-Lagrangian methods on adaptive grids based on wavelet interpolation [17,18], see also [19] for a convergence proof. Note also a hybrid method using only a semi-Lagrangian method in the velocity space [20], and a conservative semi-Lagrangian method based on WENO reconstruction [2,21]. Now somewhat outdated comparisons of different types of methods for the Vlasov equation can be found in [22,23].…”