Error estimates of the cubic interpolated pseudo-particle scheme for one-dimensional advection equations
<|reference_start|>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.<|reference_end|>
@article{kashiwabara2024error,
    title={Error estimates of the cubic interpolated pseudo-particle scheme for
  one-dimensional advection equations},
    author={Takahito Kashiwabara and Haruki Takemura},
    journal={arXiv preprint arXiv:2402.11885},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.11885},
    primaryClass={math.NA cs.NA}
}