Semi-Lagrangian difference approximations with different stability requirements

The paper demonstrates different ways of using the semi-Lagrangian approximation depending on the fulfillment of conservation laws. A one-dimensional continuity equation and a parabolic one are taken as simple methodological examples. For these equations, the principles of constructing discrete analogues are demonstrated for three different conservation laws (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 difference problems of different types as well as different ways to justify their stability.