An exactly conservative semi-Lagrangian scheme (CIP-CSL) in one dimension

Two semi-Lagrangian schemes that guarantee exactly mass conservation are pr oposed. Although they are in a nonconservative form, the interpolation func tions are constructed under the constraint of conservation of cell-integrat ed value (mass) that is advanced by remapping the Lagrangian volume. Conseq uently, the resulting schemes conserve the mass for each computational grid cell. One of them (CIP-CSL4) is the direct extension of the original cubic -interpolated propagation (CIP) method in which a cubic polynomial is used as the interpolation function and the gradient is calculated according to t he differentiated advection equation. A fourth-order polynomial is employed as the interpolation function in the CIP-CSL4 method and mass conservation is incorporated as an additional constraint on the reconstruction of the i nterpolation profile. In another scheme (CIP-CSL2), the CIP principle is ap plied to integrated mass and the interpolation function becomes quadratic. The latter one can be readily extended to multidimensions. Besides the line ar advection transportation equation, these schemes are also applied to the nonlinear advection problem with a large Courant-Freidrichs-Lewy number.