Optimal error bounds on the exponential integrator for dispersive equations with highly concentrated potential

Abstract

We study a one-dimensional linear dispersive equation of differential order $κ\geq 2$ with concentrated potential of extension $\varepsilon$ with $0 < \varepsilon \ll 1$, featuring a competition between weak dispersion of strength $\varepsilon^α\ (0 \leq α\leq κ)$ and localization induced by the concentrated potential. We first obtain precise regularity estimates of the exact solution in terms of $\varepsilon$. We then apply a natural first-order exponential integrator with step size $τ$ to discretize the equation, and establish an optimal error bound of the form $O_{L^\infty}(τ\varepsilon^β)$ (up to logarithmic factors in $τ$ and $\varepsilon$). Salient features of the result are: (i) error bounds are not only uniform in $\varepsilon$ but improve as $\varepsilon \rightarrow 0$; and (ii) no restriction on $τ$ in terms of $\varepsilon$. The analysis combines iterated Duhamel's expansions and a transformation that exploits cancellations in oscillatory phases that cannot be obtained directly from regularity estimates of the exact solution. We also show that other classical numerical schemes, such as Lie or centered splitting schemes and low regularity integrators, fail to display optimal rates of convergence. Extensive numerical results are presented and confirm the theoretical error estimates.

Figures (8)

• Figure 1.1: Plots of the solution $\mu$ at $z=8$ of various $\varepsilon$ for \ref{['eq:mu']} with $D_\kappa = D_2 = \partial_{xx}$ and $\alpha = 3/4$ (left), $\alpha = 1$ (middle), and $\alpha = 4/3$ (right)
• Figure 1.2: Plots of the numerical solution at $z=1$ obtained by the exponential integrator (left), the splitting method (middle), and the low regularity integrator (right) for \ref{['eq:schrodinger']} with $\alpha = 1$ and $\varepsilon = 2^{-10}$
• Figure 5.1: The difference of the exact solution $\mu(z)$ and the free solution $e^{iz\varepsilon^\alpha}\mu_0$ at $z = 1$ in terms of $\varepsilon$ for $\alpha = \frac{3}{4}$ (left), $\alpha = 1$ (middle), and $\alpha = \frac{4}{3}$ (right)
• Figure 5.2: The difference of the exact solution $\mu(z)$ and the free solution $e^{iz\varepsilon^\alpha}\mu_0$ at $z = 1$ in terms of $\varepsilon$ for $\alpha = 1$ (left), $\alpha = \frac{3}{2}$ (middle), and $\alpha = 2$ (right)
• Figure 5.3: Errors of the exponential integrator divided by $\min\{\varepsilon^{1+\frac{\alpha}{2}}, \varepsilon^{2-\alpha} \ln \varepsilon^{-1}\}$ with $\alpha = \frac{1}{2}$ (left), $\alpha = \frac{2}{3}$ (middle), and $\alpha = 1$ (right) for \ref{['eq:mu']} with $D_\kappa = \partial_x^2$

Theorems & Definitions (34)

• Theorem 2.1
• Remark 1
• Remark 2
• Theorem 2.2
• Remark 3
• Remark 4
• Lemma 1
• Lemma 2
• proof
• Lemma 3