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arXiv · 2512.02390

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

Haruki Takemura
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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 (Δt^r + h^{2s} / Δt) $ and an $ H^s $-error estimate of $ O (Δt^r + h^{s} / \sqrt{Δt}) $, where $ h $ and $ Δt $ 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 $Δt$, and it reduces to the $L^2$-norm in the limit $ h \to 0 $.

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Haruki Takemura. 2025-12-02. Error estimates for semi-Lagrangian schemes with higher-order interpolation for conservation laws with dispersive terms. https://arxiv.org/abs/2512.02390

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