Optimal error bounds on the exponential integrator for dispersive equations with highly concentrated potential
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911468443664384 |
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| author | Bal, Guillaume Wang, Chushan |
| author_facet | Bal, Guillaume Wang, Chushan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_22068 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Optimal error bounds on the exponential integrator for dispersive equations with highly concentrated potential Bal, Guillaume Wang, Chushan Numerical Analysis 65M15, 35Q55, 65M70, 35R60 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. |
| title | Optimal error bounds on the exponential integrator for dispersive equations with highly concentrated potential |
| topic | Numerical Analysis 65M15, 35Q55, 65M70, 35R60 |
| url | https://arxiv.org/abs/2602.22068 |