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.

翻译：本文针对周期性边界条件下的一维平流方程，提出了三次插值伪粒子格式（CIP格式）的误差估计。CIP格式是一种采用分段三次Hermite插值的半拉格朗日方法。尽管数值实验已知该格式具有三阶时空精度，但其严格证明仍是一个未解决的问题。本文中，记空间与时间步长分别为$h$和$\Delta t$，我们在理论上证明了$L^2$范数下的误差估计为$O(\Delta t^3 + \frac{h^4}{\Delta t})$，该结果在$h = O(\Delta t)$条件下验证了前述数值预测。证明基于插值算子的性质，其中最关键的性质是：该算子对二阶导数的行为相当于$L^2$投影算子。我们指出，相同策略同样适用于分析采用三次样条插值的半拉格朗日方法的误差估计。