物理问题中高指标Sturm-Liouville特征值的高效计算
Efficient computation of high index Sturm-Liouville eigenvalues for problems in physics
AI总结：
针对高指标Sturm-Liouville特征值计算中解的高度振荡导致数值步长受限的问题，提出基于系数逼近的（修正）Magnus或Neumann积分器方法，实现全谱均匀近似并允许大步长计算。
摘要：
求解Sturm-Liouville问题的特征值可能是一项计算上具有挑战性的任务，尤其是当需要计算大量特征值或仅寻求特别大的特征值时。这是由于高特征值对应的解具有高度振荡行为，迫使简单的积分器采取越来越小的步长。我们将讨论一些技术，这些技术在整个特征值谱上产生均匀近似，并且即使对于高特征值也能采取大步长。特别地，我们将重点介绍基于系数逼近的方法，该方法将Sturm-Liouville问题的系数函数替换为更简单的近似，然后求解近似问题。使用（修正的）Magnus或Neumann积分器可以将系数逼近思想扩展到高阶方法。
英文摘要：
Finding the eigenvalues of a Sturm-Liouville problem can be a computationally challenging task, especially when a large set of eigenvalues is computed, or just when particularly large eigenvalues are sought. This is a consequence of the highly oscillatory behaviour of the solutions corresponding to high eigenvalues, which forces a naive integrator to take increasingly smaller steps. We will discuss some techniques that yield uniform approximation over the whole eigenvalue spectrum and can take large steps even for high eigenvalues. In particular, we will focus on methods based on coefficient approximation which replace the coefficient functions of the Sturm-Liouville problem by simpler approximations and then solve the approximating problem. The use of (modified) Magnus or Neumann integrators allows to extend the coefficient approximation idea to higher order methods.
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