We propose a new class of uniformly accurate splitting methods for the Benjamin-Bona-Mahony equation which converge uniformly in the dispersive parameter $\varepsilon$. The proposed splitting schemes are furthermore asymptotic convergent and preserve the KdV limit. We carry out a rigorous convergence analysis of the splitting schemes exploiting the smoothing properties in the system. This will allow us to establish improved error bounds with gain either in regularity (for non smooth solutions) or in the dispersive parameter $\varepsilon$. The latter will be interesting in regimes of a small dispersive parameter. We will in particular show that in the classical BBM case $P(\partial_x) = \partial_x$ our Lie splitting does not require any spatial regularity, i.e, first order time convergence holds in $H^{r}$ for solutions in $H^{r}$ without any loss of derivative. This estimate holds uniformly in $\varepsilon$. In regularizing regimes $\varepsilon=\mathcal{O}(1) $ we even gain a derivative with our time discretisation at the cost of loosing in terms of $\frac{1}{\varepsilon}$. Numerical experiments underline our theoretical findings.
翻译:我们为Benjamin-Bona-Bona-Mahony等式建议了一个新的统一准确的分解方法类别,该类方法在分散参数$\varepsilon$中统一一致。提议的分解方案进一步零散,并保留KdV的界限。我们对利用系统中平滑特性的分解方案进行了严格的趋同分析。这将使我们能够在正常(不顺利的解决办法)或分散参数$\varepsilon$中建立更好的差错界限。在分散参数中,后者对一个小分散参数的体系将很感兴趣。我们特别要表明,在经典的BBM案 $(部分_x) =\部分_x) =\部分_x$ 我们的分解并不要求任何空间规律性,也就是说,在不损耗损任何衍生物的$H*r} 中,第一个顺序趋同时间界限以美元维持。这一估计以$\varepsilonlon$统一。在固定制度中,在小型分散参数制度中将很感兴趣。我们甚至以1美元获得我们时间分化的模型分析结果。