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

TL;DR

This paper provides error estimates for semi-Lagrangian schemes with higher-order interpolation applied to one-dimensional dispersive conservation laws, including the Korteweg--de Vries equation, demonstrating stability and convergence properties.

Contribution

It introduces new error bounds for semi-Lagrangian schemes with spline or Hermite interpolation for dispersive conservation laws, extending the theoretical understanding of their accuracy and stability.

Findings

Derived $L^2$ and $H^s$ error estimates for the schemes.

Proved stability of higher-order interpolation operators in relevant norms.

Established conditions under which the schemes are stable and accurate.

Abstract

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 $ 2 s - 1 $, we derive an $L^2$-error estimate of $ O (\Delta t^r + h^{2s} / \Delta t) $ and an $ H^s $-error estimate of $ O (\Delta t^r + h^{s} / \sqrt{\Delta t}) $, where $ h $ and $ \Delta t $ denote the spatial mesh size and the time step size, respectively, and $ r \in \lparen 0, 1\rbrack $ 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 $\Delta t$, and it reduces to the $L^2$-norm in the limit $ h \to 0 $.