Vlasov simulation/semi-Lagrangian dG
Performing Vlasov simulation (kinetic description of a plasma) poses many challenges for numerical methods. In addition to the large dimensional phase space, the system suffers from a restrictive CFL condition and it is vital to respect certain physical properties of the flow. Particle in cell (PIC) methods have been, and still are, heavily used. However, certain physical phenomena are not well resolved by particle methods (e.g. Landau damping or problems in which the tail of the density function is important).
Semi-Lagrangian schemes have emerged as a viable alternative to efficiently solve such problems. However, many of the well-established methods (such as spline interpolation or spectral methods) are ill-suited for modern computer architectures due to their global data dependency. Semi-Lagrangian discontinuous Galerkin methods are completely local, conservative by construction, and limit the amount of diffusion that is introduced into the system.
Semi-Lagrangian discontinuous Galerkin methods compare very favorable to cubic spline interpolation and have excellent performance on GPUs. The developed open source SLDG code can be used to perform kinetic simulation using this method. We have also published a rigorous convergence analysis of this approach.