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

• Published in: Journal of computational physics 2014-08, Vol.270, p.711-752
• Main Authors: Güçlü, Yaman, Christlieb, Andrew J., Hitchon, William N.G.
• Format: Article
• Language: English
• Subjects: Charged particles; Convected Scheme; Dynamical systems; Mathematical models; Nonlinear dynamics; Nonlinearity; Semi-Lagrangian; Spectrally accurate; Trajectories; Transport equations; Velocity distribution; Vlasov–Poisson
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2014.04.003

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 (Güçlü and Hitchon, 2012 [1]) 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.