Conservative Semi-Lagrangian Finite Difference WENO Formulations with Applications to the Vlasov Equation (Communications in Computational Physics, 2011)
In this paper, we propose a new conservative semi-Lagrangian (SL) finitedifference (FD) WENO scheme for linear advection equations, which can serve as abase scheme for the Vlasov equation by Strang splitting [4]. The reconstruction procedurein 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 approximatethe integral form of the equation in a FV scheme, we input point values and approximatethe 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 shearingin velocity (v) and electrical field (E) over a cell when performing the Strang splittingto 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 forsolving the Vlasov-Poisson (VP) system. We perform numerical experiments on linearadvection, rigid body rotation problem; and on the Landau damping and two-streaminstabilities by solving the VP system. For comparison, we also apply (1) the conservativeSL FD WENO scheme, proposed in [22] for incompressible advection problem, (2)the conservative SL FD WENO scheme proposed in [21] and (3) the non-conservativeversion of the SL FD WENO scheme in [3] to the same test problems. The performancesof 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.