We study necessary conditions for stability of a Numerov-type compact higher-order finite-difference scheme for the 1D homogeneous wave equation in the case of non-uniform spatial meshes. We first show that the uniform in time stability cannot be valid in any spatial norm provided that the complex eigenvalues appear in the associated mesh eigenvalue problem. Moreover, we prove that then the solution norm grows exponentially in time making the scheme strongly non-dissipative and therefore impractical. Numerical results confirm this conclusion. In addition, for some sequences of refining spatial meshes, an excessively strong condition between steps in time and space is necessary (even for the non-uniform in time stability) which is familiar for explicit schemes in the parabolic case.

我们研究了非均匀空间网格情况下一维齐次波动方程Numerov型紧凑型高阶有限差分格式稳定性的必要条件。我们首先证明，只要复杂特征值出现在相关的网格特征值问题中，时间稳定性就不可能在任何空间范数中都是有效的。此外，我们证明了，解范数随时间呈指数增长，使得该方案非常耗散，因此不切实际。数值结果证实了这一结论。另外，对于某些细化空间网格的序列，时间和空间步长之间的条件过强是必要的（即使对于时间稳定性而言也是不均匀的），这对于抛物线情况下的显式方案是很熟悉的。