Upper and Lower Error Bounds for a Compact Fourth-Order Finite-Difference Scheme for the Wave Equation with Nonsmooth Data

The compact fourth-order finite-difference scheme for solving the 1d wave equation is studied. New error bounds of the fractional order $\mathcal{O}(h^{4(\lambda-1)/5})$ are proved in the mesh energy norm in terms of data, for two initial functions from the Sobolev and Nikolskii spaces with the smoothness order $\lambda$ and $\lambda-1$ and the free term with a dominated mixed smoothness of order $\lambda-1$, for $1\leq\lambda\leq 6$. The corresponding lower error bounds are proved as well to ensure the sharpness in order of the above error bounds with respect to each of the initial functions and the free term for any $\lambda$. Moreover, they demonstrate that the upper error bounds cannot be improved if the Lebesgue summability indices in the error norm are weakened down to 1 both in $x$ and $t$ and simultaneously the summability indices in the norms of data are strengthened up to $\infty$ both in $x$ and $t$. Numerical experiments confirming the sharpness of the mentioned orders for half-integer $\lambda$ and piecewise polynomial data have already been carried out previously.