We consider compact finite-difference schemes of the 4th approximation order for an initial-boundary value problem (IBVP) for the $n$-dimensional non-homogeneous wave equation, $n\geq 1$. Their construction is accomplished by both the classical Numerov approach and alternative technique based on averaging of the equation, together with further necessary improvements of the arising scheme for $n\geq 2$. The alternative technique is applicable to other types of PDEs including parabolic and time-dependent Schr\"{o}dinger ones. The schemes are implicit and three-point in each spatial direction and time and include a scheme with a splitting operator for $n\geq 2$. For $n=1$ and the mesh on characteristics, the 4th order scheme becomes explicit and close to an exact four-point scheme. We present a conditional stability theorem covering the cases of stability in strong and weak energy norms with respect to both initial functions and free term in the equation. Its corollary ensures the 4th order error bound in the case of smooth solutions to the IBVP. The main schemes are generalized for non-uniform rectangular meshes. We also give results of numerical experiments showing the sensitive dependence of the error orders in three norms on the weak smoothness order of the initial functions and free term and essential advantages over the 2nd approximation order schemes in the non-smooth case as well.
翻译:我们认为,第4号近似第4号近似顺序对于初始界限值问题(IBVP)的缩略式有限差异方案(IBVP)对美元和非同质波方程式($n\geq 1$)都是隐含的和三点的,包括一个以美元=qe2美元为单位的分裂操作器($n\geq 2美元)和基于等式平均法的替代技术的构建。同时,对正在形成的美元2美元办法作了进一步的必要改进。替代技术适用于其他种类的PDE,包括抛线和时间依赖的Schr\{o}dinger。这些办法在每个空间方向和时间都是隐含的和三点的,并包括一个以美元=qe2美元为单位的分裂操作器。第4号命令方案明确和接近一个精确的四点方案。我们提出了一个有条件的稳定性词,涵盖了在初始功能和自由期限两方面的强弱能能源规范下的稳定情况。其必然确保了IBVPP的平滑式解决方案中包含的第四顺序错误。在非正统办法中,在非正态性初步规则中,主要的平稳规则是表明不平稳的不平稳规则。