论文标题
波动方程的紧凑型高阶有限差分方案在不均匀的网格上可能是强烈的
A compact higher-order finite-difference scheme for the wave equation can be strongly non-dissipative on non-uniform meshes
论文作者
论文摘要
我们研究了在不均匀的空间网格的情况下，对于1D均匀波方程的Numerov型紧凑型高阶差异方案的稳定性的必要条件。我们首先表明，如果复杂的特征值出现在相关的网格特征值问题中，那么时间稳定性在任何空间规范中都不能有效。此外，我们证明，解决方案规范会及时地呈指数增长，从而使该方案强烈非疾病，因此不切实际。数值结果证实了这一结论。此外，对于一些精炼空间网格的序列，需要在时间和空间的步骤之间过度强烈的条件（即使对于时间稳定性的不均匀），这对于抛物线层中的显式方案很熟悉。
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.
