Error estimates of the cubic interpolated pseudo-particle scheme for one-dimensional advection equations
Takahito Kashiwabara, Haruki Takemura
TL;DR
The paper delivers a rigorous $L^2$-norm error bound for the cubic interpolated pseudo-particle (CIP) scheme solving the 1D advection equation with periodic boundaries, proving $||\varphi^n-\varphi_h^n||_{L^2} = O(\Delta t^3 + \frac{h^4}{\Delta t})$ under suitable regularity and mesh assumptions. A weighted $H^2$ norm is introduced to overcome the unboundedness of cubic Hermite interpolation in $L^2$, enabling stability and convergence proofs; the analysis shows the interpolation acts like an $L^2$ projection on second derivatives. The results extend to a semi-Lagrangian method using cubic spline interpolation, yielding comparable convergence bounds and substantiating the CIP approach on nonuniform meshes. Numerical experiments corroborate the theoretical rates, reveal near-3rd-order accuracy in practice, and highlight CIP’s superior phase accuracy for high-frequency modes compared to spline-based or upwind schemes. Overall, the work justifies the third-order space-time accuracy of CIP and provides a robust framework for error control in semi-Lagrangian advection solvers.
Abstract
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 $ Δt $ respectively, we prove an error estimate $ O(Δt^3 + \frac{h^4}{Δt}) $ in $ L^2 $ norm theoretically, which justifies the above-mentioned prediction if $ h = O(Δt) $. 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.