Arbitrarily high order Convected Scheme solution of the Vlasov–Poisson system

Abstract

The Convected Scheme (CS) is a 'forward-trajectory' semi-Lagrangian method for solution of transport equations, which has been most often applied to the kinetic description of plasmas and rarefied neutral gases. In its simplest form, the CS propagates the solution forward in time by advecting the so called 'moving cells' along their characteristic trajectories, and by remapping them on the mesh at the end of the time step. The CS is conservative, positivity preserving, simple to implement, and it is not subject to time step restriction to maintain stability. Recently [Y. Güçlü, W. N. G. Hitchon, A high order cell-centered semi-Lagrangian scheme for multi-dimensional kinetic simulations of neutral gas flows, J. Comput. Phys. 231 (8) (2012) 3289-3316] a new methodology was introduced for reducing numerical diffusion, based on a modified equation analysis: the remapping error was compensated by applying small corrections to the final position of the moving cells prior to remapping. While the spatial accuracy was increased from 2nd to 4th order, the new scheme retained the important properties of the original method, and was shown to be extremely simple and efficient for constant advection problems.Here the CS is applied to the solution of the Vlasov-Poisson system, which describes the evolution of the velocity distribution function of a collection of charged particles subject to reciprocal Coulomb interactions. The Vlasov equation is split into two constant advection equations, one in configuration space and one in velocity space, and high order time accuracy is achieved by proper composition of the operators. The splitting procedure enables us to use the constant advection solver, which we extend to arbitrarily high order of accuracy in time and space: a new improved procedure is given, which makes the calculation of the corrections straightforward. Focusing on periodic domains, we describe a spectrally accurate scheme based on the fast Fourier transform; the proposed implementation is strictly conservative and positivity preserving. The ability to correctly reproduce the system dynamics, as well as resolving small-scale features in the solution, is shown in classical 1D-1V test cases, both in the linear and the non-linear regimes.

ColDICE: A parallel Vlasov–Poisson solver using moving adaptive simplicial tessellation

“…Many other grid based Vlasov solvers have been proposed since the seminal contribution of Cheng and Knorr, most of them being of semi-Lagrangian nature [see, e.g., 131,130,150,142,115,132,62,11,23,7,141,50,49,124,40,128,78,73, but this list is far from being exhaustive]. For instance one can mention the recent Vlasov-Poisson simulations of Yoshikawa, Yoshida & Umemura [149] in six-dimensional phase-space using the positive flux conservation scheme [62].…”

An Explicit High-Order Single-Stage Single-Step Positivity-Preserving Finite Difference WENO Method for the Compressible Euler Equations

“…In [48], a flux cut-off limiter is also applied to FD WENO schemes to retain positivity. In addition to gas dynamics, plasma physics is another area where retaining positivity of numerical solutions is critical, and therefore has seen recent attention in the literature [49,50]. For example, collision operators for Vlasov equations require a positive distribution function in order to avoid creating artificial singularities.…”

Solving hyperbolic-elliptic problems on singular mapped disk-like domains with the method of characteristics and spline finite elements

“…with s s :" ps``s´q{2 and d :" ps`´s´q{2. If the smoothing layer is small enough, we can still rely on the analytical result obtained for the dispersion relation in the case of the sharp annular layer (33). The numerical results have been verified against the analytical dispersion relation for a perturbation with azimuthal mode number m " 9 and amplitude " 10´4.…”