Conservative Semi-Lagrangian Methods for Kinetic Equations

The chapter is a review on semi-Lagrangian methods for kinetic equations. In semi-Lagrangian schemes for Boltzmann and BGK models, the equation for the distribution function is integrated along the characteristics, and the value of the unknown is computed on grid nodes in phase space. The value at the foot of the characteristic is computed by suitable interpolation. This approach allows the use of much larger time steps, compared to a Eulerian treatment of the transport term, since the latter requires CFL-type limitations on the time step, thus resulting in more efficient schemes. The use of standard non-oscillatory reconstructions may cause lack of conservation of semi-Lagrangian schemes. Conservation is restored by adopting a suitable conservative reconstruction, which allows at the same time high-order approximation and lack of spurious oscillations. Constrained \(L^2\) -minimization is adopted to impose conservation also at the level of the discrete collisional term. The technique is general and can be applied to a variety of kinetic equations, including BGK model for single gas and for mixtures, Boltzmann equation for rarefied gas dynamics, and Vlasov-Poisson systems. In the latter case the conservative reconstruction can be directly applied by using splitting methods, while non-splitting methods require a slightly different treatment. The use of high-order reconstruction and high-order time integrators, together with spectral methods in velocity, allows very accurate schemes for the numerical solution of collisional kinetic equations. Finally, an account of the theoretical results on the convergence properties of the method is provided at the end of the chapter.