论文标题
一种自适应光谱方法，用于振荡性二阶线性ODE，频率无关的成本
An adaptive spectral method for oscillatory second-order linear ODEs with frequency-independent cost
论文作者
论文摘要
我们引入了一种有效的数值方法，用于二阶线性ODE，其解决方案可能会在高度振荡和在解决方案间隔内缓慢变化之间变化。在振荡区域中，该溶液是通过遵守非线性riccati方程的非振荡相函数生成的。我们提出了一种缺陷校正迭代，为这种相位函数提供了渐近系列。这在数字上是在Chebyshev网格上近似的，并具有少量的节点。对于分析系数，我们证明每次迭代均达到一定的最大数量，将残差降低到局部频率的一定级数。该算法可以适应方法的步骤大小和选择，并切换为远离振荡区域的常规光谱搭配方法。在数值实验中，我们发现我们的提案的表现优于其他最先进的振荡求解器，最明显的是，在低到中等频率下，在低公差下，它可能会使用最多$ 10^6美元的功能评估。即使在高频制度下，我们的实施平均比其他专业求解器快10倍。
We introduce an efficient numerical method for second order linear ODEs whose solution may vary between highly oscillatory and slowly changing over the solution interval. In oscillatory regions the solution is generated via a nonoscillatory phase function that obeys the nonlinear Riccati equation. We propose a defect-correction iteration that gives an asymptotic series for such a phase function; this is numerically approximated on a Chebyshev grid with a small number of nodes. For analytic coefficients we prove that each iteration, up to a certain maximum number, reduces the residual by a factor of order of the local frequency. The algorithm adapts both the step size and the choice of method, switching to a conventional spectral collocation method away from oscillatory regions. In numerical experiments we find that our proposal outperforms other state-of-the-art oscillatory solvers, most significantly at low-to-intermediate frequencies and at low tolerances, where it may use up to $10^6$ times fewer function evaluations. Even in high frequency regimes, our implementation is on average 10 times faster than other specialized solvers.
