短波-长波相互作用的多尺度分辨率
Multiscale Resolution of Shortwave-Longwave Interaction
AI总结：
针对高频短波与低频长波混合的波动问题，提出一种多尺度算法，以降低计算成本。
摘要：
在时间相关波的研究中，求解高频（短波，波数k=kmax）和低频（长波，接近k=kmin）混合的问题在计算上是昂贵的。考虑一个低频波在尖锐杂质上散射的问题。杂质产生高频波，这些波传播并扩散到整个计算域，而计算域必须足够大以包含几个长波。传统的谱方法的计算成本与O(kmax/kmin log(kmax/kmin))成正比。我们在这里提出一种多尺度算法（针对薛定谔方程实现，但普遍适用），该算法以空间和时间上的O(kmax L log(kmax / kmin) log(kmax L))成本解决问题。这里，L是算法解析所有频率的区域的宽度，且与kmin无关。
英文摘要：
In the study of time-dependent waves, it is computationally expensive to solve a problem in which high frequencies (shortwaves, with wavenumber k = kmax) and low frequencies (longwaves, near k=kmin) mix. Consider a problem in which low frequencies scatter off a sharp impurity. The impurity generates high frequencies which propagate and spread throughout the computational domain, while the domain must be large enough to contain several longwaves. Conventional spectral methods have computational cost proportional to O(kmax/kmin \log (kmax/kmin)). We present here a multiscale algorithm (implemented for the Schrodinger equation, but generally applicable) which solves the problem with cost (in space and time) O(kmax L log(kmax / kmin) \log(kmax L)). Here, L is the width of the region in which the algorithm resolves all frequencies, and is independent of kmin.
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