Error estimates for semi-Lagrangian schemes with higher-order interpolation for conservation laws with dispersive terms
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908687012986880 |
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| author | Takemura, Haruki |
| author_facet | Takemura, Haruki |
| contents | 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 $. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_02390 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Error estimates for semi-Lagrangian schemes with higher-order interpolation for conservation laws with dispersive terms Takemura, Haruki Numerical Analysis Primary: 65M12, Secondary: 65M06, 65M25 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 $. |
| title | Error estimates for semi-Lagrangian schemes with higher-order interpolation for conservation laws with dispersive terms |
| topic | Numerical Analysis Primary: 65M12, Secondary: 65M06, 65M25 |
| url | https://arxiv.org/abs/2512.02390 |