Abstract: Mathematical study on semi_Lagrangian advection computational scheme is carried out. Firstly, by the characteristic curves of the partial differential equation of first order, the existence and the uniqueness of the characteristic curves are proven. Secondly, by Picard's and В. Канторович's mechods, the search for the departure point is sloved systematically. At the end, from differential and computational geometry, one_dimensional shape perserving interpolation on the basis of non_linear spline is proposed, and two dimensional shape perserving interpolation on the basis of Lagrangian, bicubic spline and Coon's surfaces is introduced.

Key words: characteristic curves, departure point, shape-presverving interpolation