Abstract
In this chapter we will introduce different techniques that have been developed to ensure that the semi-Lagrangian scheme can preserve the shape of the body being advected, to avoid the Gibbs phenomena of the under- and overshoots. We shall introduce the cascade interpolation methods between Eulerian and Lagrangian grids, along with different finite volume approaches, that enable the conservation of mass either locally of globally. We will also introduce SLICE which is a finite volume based scheme that utilizes the cascade interpolation. The chapter is finished by introducing the Vlasov family of nonlinear PDEs and the HWENO approach to solve them.
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