Error estimates for semi-Lagrangian schemes with higher-order interpolation for conservation laws with dispersive terms

We establish error estimates for semi-Lagrangian schemes for the initial value problem of one-dimensional conservation laws with a dispersive term, including the Korteweg–de Vries equation. The schemes considered in this paper are based on the semi-Lagrangian technique combined with spatial discretization by higher-order interpolation operators. For the semi-Lagrangian schemes equipped with the spline or Hermite interpolation operators of order $2s-1$, we derive an $L^2$-error estimate of $O(\mathrm{\Delta }{t}^r+h^{2s}\mathrm{/}\mathrm{\Delta }t)$ and an $H^s$-error estimate of $O(\mathrm{\Delta }{t}^r+h^s\mathrm{/}\sqrt{\mathrm{\Delta }t})$, where $h$ and $\mathrm{\Delta }t$ denote the spatial mesh size and the time step size, respectively, and $r\in (0,1]$ is a parameter determined by the discretization of the dispersive term. A key step in the analysis is to establish the stability of the interpolation operators. Under suitable assumptions, interpolation operators of order $2s-1$ are stable with respect to the $H^s$-norm as well as a weighted $H^s$-norm. The weighted $H^s$-norm depends on $h$ and $\mathrm{\Delta }t$, and it reduces to the $L^2$-norm in the limit $h\to 0$.