  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.
