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

Haruki Takemura

TL;DR

This work analyzes fully implicit semi-Lagrangian schemes for dispersive conservation laws, including KdV, using higher-order spline and Hermite interpolation. It derives L^2 and H^s error bounds that combine time stepping and spatial interpolation errors, fundamentally relying on stability properties of the interpolation operators in both standard and weighted Sobolev norms. The analysis shows the error scales as Δt^r + h^{q-s}/Δt^{1/2} in the H^s framework (and Δt^r + h^{q}/Δt in the weighted L^2 framework), with r determined by the dispersive discretization (r = 1/3 for Λ4 and r = 2/3 for Λ5). Numerical experiments corroborate the theoretical rates, demonstrating the practical effectiveness of the approach for large-time, high-gradient dispersive flows.

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 $.

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

TL;DR

This work analyzes fully implicit semi-Lagrangian schemes for dispersive conservation laws, including KdV, using higher-order spline and Hermite interpolation. It derives L^2 and H^s error bounds that combine time stepping and spatial interpolation errors, fundamentally relying on stability properties of the interpolation operators in both standard and weighted Sobolev norms. The analysis shows the error scales as Δt^r + h^{q-s}/Δt^{1/2} in the H^s framework (and Δt^r + h^{q}/Δt in the weighted L^2 framework), with r determined by the dispersive discretization (r = 1/3 for Λ4 and r = 2/3 for Λ5). Numerical experiments corroborate the theoretical rates, demonstrating the practical effectiveness of the approach for large-time, high-gradient dispersive flows.

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 , we derive an -error estimate of and an -error estimate of , where and denote the spatial mesh size and the time step size, respectively, and 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 are stable with respect to the -norm as well as a weighted -norm. The weighted -norm depends on and , and it reduces to the -norm in the limit .

Paper Structure

This paper contains 13 sections, 7 theorems, 135 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Reformulation of the fully semi-Lagrangian scheme
  3. Derivation of the scheme
  4. Reformulation of the scheme
  5. Main results
  6. Key lemmas and the proofs of the main theorems
  7. Preparations
  8. Consistency
  9. Proofs of the main theorems
  10. Step 1.
  11. Step 2.
  12. Step 3.
  13. Numerical experiments

Key Result

Lemma 2.1

Let $s \geq 2$ be an integer. Assume that $v \in H^s (\mathbb{T})$ and $f \in W^{s, \infty} (\mathbb{T})$. Suppose that is satisfied. Then the following three statements hold:

Figures (2)

  • Figure 1: Relative $L^2$-error versus the time step $\Delta t$. The mesh is uniform in both space and time, and the spatial mesh size is fixed at $h = 1 / 1000$. The $L^2$-errors are evaluated at $T = 1$.
  • Figure 2: Relative $L^2$-error versus the spatial mesh size $h$. The mesh is uniform in both space and time. The time step size is given by $\Delta t = 100 h^{12/5}$. The $L^2$-norm is evaluated at $T = N_t \Delta t$, where $N_t = \lfloor 1 / \Delta t \rfloor$.

Theorems & Definitions (10)

  • Lemma 2.1: Lemmas 2.1 and A.1 of 25Ta
  • Theorem 3.1
  • Theorem 3.2
  • Lemma 4.1: Lemma A.3 of 25Ta
  • Lemma 4.2: Lemma A.4 of 25Ta
  • Lemma 4.3
  • Lemma 4.4
  • proof
  • proof : Proof of Theorem \ref{['thm:s']}
  • proof : Proof of Theorem \ref{['thm:2']}