Conservative and positivity-preserving semi-Lagrangian kinetic schemes with spectrally accurate phase-space resolution

Summary form only given. The Convected Scheme (CS) is a family of semi-Lagrangian algorithms, most usually applied to the solution of Boltzmann's equation, which uses a method of characteristics in an integral form to project a moving cell (MC) forward to a group of mesh cells. In earlier work [1], a 4th-order version of the cell-centered CS was presented, which was based on applying an a-priori correction to the position of the MC after the ballistic move and prior to remapping to the mesh. Such corrections were calculated by means of a modified equation analysis applied to the continuity equation with a prescribed flow field. The resulting 4th-order CS showed a drastically reduced numerical diffusion, while it retained the desirable properties of the original scheme (i.e. mass conservation, positivity preservation, and simplicity). In this contribution we describe higher order versions of the CS, suited to the accurate solution of the Vlasov equation with minimum computational resources. By applying an appropriate operator splitting procedure, the solution to the Vlasov-Poisson (or Vlasov-Maxwell) system can be reduced to a succession of constant advection steps, either in configuration or in velocity space, interleaved with appropriate field updates. With this setting in mind, we specialize our analysis to the constant advection equation, and we illustrate a new procedure that extends the CS to arbitrarily high order of accuracy. We describe a nominally 22nd-order CS, in which we compute the required 20 spatial derivatives of the solution using a fast Fourier transform. For smooth profiles, this scheme shows spectral convergence to the exact solution, and hence we refer to it as “Spectral CS”. Further, adaptive filtering in Fourier space permits us to resolve non smooth profiles without introducing spurious oscillations. We show the scheme's behavior in typical 1D-1V test cases for the Vlasov-Poisson system, both in periodic and bounded domains- with one or more species. We then discuss higher dimensional problems, where the computational savings inherent to the Spectral CS would enable unprecedented phase-space resolution. Finally we consider the solution of the Vlasov-Maxwell system, as well as the inclusion of collisional processes.