A SHAPE-PRESERVING INTERPOLATION - APPLICATIONS TO SEMI-LAGRANGIAN ADVECTION

A high-order interpolation scheme, to be applied in semi-Lagrangian advection algorithms, is discussed. An interpolation polynomial is constructed on a four-point discretization stencil and is then coupled with shape-preserving derivative estimates at the internal mesh points. The obtained interpolate of the advected profile is utilized for integration of a scalar function along the wind trajectories. The discrete maximum principle technique is applied to formulate the positivity conditions of the numerical scheme. Results of computational examples are presented for one- and two-dimensional Lagrangian advection of standard test shapes.