Error estimates of the cubic interpolated pseudo-particle scheme for one-dimensional advection equations

Error estimates of cubic interpolated pseudo-particle scheme (CIP scheme) for the one-dimensional advection equation with periodic boundary conditions are presented. The CIP scheme is a semi-Lagrangian method involving the piecewise cubic Hermite interpolation. Although it is numerically known that the space-time accuracy of the scheme is third order, its rigorous proof remains an open problem. In this paper, denoting the spatial and temporal mesh sizes by $h$ and $\Delta t$ respectively, we prove an error estimate $O(\Delta t^3 + \frac{h^4}{\Delta t})$ in $L^2$ norm theoretically, which justifies the above-mentioned prediction if $h = O(\Delta t)$. The proof is based on properties of the interpolation operator; the most important one is that it behaves as the $L^2$ projection for the second-order derivatives. We remark that the same strategy perfectly works as well to address an error estimate for the semi-Lagrangian method with the cubic spline interpolation.