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：本文给出了三次插值伪粒子格式（CIP格式）在一维周期边界对流方程中的误差估计。该格式是一种基于分段三次埃尔米特插值的半拉格朗日方法。尽管数值实验表明该格式具有三阶时空精度，但其严格证明仍是一个未解问题。本文中，记空间网格尺寸为$ h $，时间步长为$ \Delta t $，我们理论上证明了$ L^2 $范数下的误差估计$ O(\Delta t^3 + \frac{h^4}{\Delta t}) $，当$ h = O(\Delta t) $时，该结果验证了前述预测。证明基于插值算子的性质，其中最关键的一点是该算子对二阶导数具有$ L^2 $投影的行为。值得注意的是，相同策略可完美适用于采用三次样条插值的半拉格朗日方法的误差估计。