Semi-Lagrangian Difference Approximations for Distinct Transfer Operators

The paper gives a review of using the semi-Lagrangian approximation depending on the fulfillment of conservation laws for the transfer operator. We begin with approximations of the one-dimensional transfer equation and a parabolic one as simple methodological examples. For two-dimensional problems, first we apply one-dimensional approximations in two directions separately. Then we present another combined approximation along trajectories of the transfer operator. For parabolic and transfer equations, the principles of constructing discrete analogues are demonstrated for three different conservation laws of transfer operator (or the requirements of stability in the related discrete norms similar to the L-1, L-2, L-infinity - norms). It is significant that different conservation laws yield distinct difference problems as well as distinct ways to justify their stability.