An Exactly Conservative Semi-Lagrangian Scheme (CIP–CSL) in One Dimension

Abstract

Two semi-Lagrangian schemes that guarantee exactly mass conservation are proposed. Although they are in a nonconservative form, the interpolation functions are constructed under the constraint of conservation of cell-integrated value (mass) that is advanced by remapping the Lagrangian volume. Consequently, 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 the 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 interpolation profile. In another scheme (CIP–CSL2), the CIP principle is applied to integrated mass and the interpolation function becomes quadratic. The latter one can be readily extended to multidimensions. Besides the linear advection transportation equation, these schemes are also applied to the nonlinear advection problem with a large Courant–Freidrichs–Lewy number.