SIAM Journal on Scientific Computing, Volume 43, Issue 5, Page A3244-A3268, January 2021.
When using Laguerre and Hermite spectral methods to numerically solve PDEs in unbounded domains, the number of collocation points assigned inside the region of interest is often insufficient, particularly when the region is expanded or translated in order to safely capture the unknown solution. Simply increasing the number of collocation points cannot ensure a fast convergence to spectral accuracy. In this paper, we propose a scaling technique and a moving technique to adaptively cluster enough collocation points in a region of interest in order to achieve fast spectral convergence. Our scaling algorithm employs an indicator in the frequency domain that both is used to determine when scaling is needed and informs the tuning of a scaling factor to redistribute collocation points in order to adapt to the diffusive behavior of the solution. Our moving technique adopts an exterior-error indicator and moves the collocation points to capture the translation. Both frequency and exterior-error indicators are defined using only the numerical solutions. We apply our methods to a number of different models, including diffusive and moving Fermi--Dirac distributions and nonlinear Dirac solitary waves, and demonstrate recovery of spectral convergence for time-dependent simulations. A performance comparison in solving a linear parabolic problem shows that our frequency scaling algorithm outperforms the existing scaling approaches. We also show our frequency scaling technique is able to track the blowup of average cell sizes in a model for cell proliferation. In addition to the Laguerre and Hermite basis functions with exponential decay at infinity, we also successfully apply the frequency-dependent scaling technique into rational basis functions with algebraic decay at infinity.

中文翻译：

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无界域中谱方法的有效缩放和移动技术

SIAM 科学计算期刊，第 43 卷，第 5 期，第 A3244-A3268 页，2021 年 1 月。
当使用 Laguerre 和 Hermite 谱方法对无界域中的偏微分方程进行数值求解时，在感兴趣区域内分配的搭配点数量通常不足，特别是当区域被扩展或平移以安全捕获未知解时。简单地增加搭配点的数量并不能确保快速收敛到光谱精度。在本文中，我们提出了一种缩放技术和一种移动技术，以在感兴趣的区域中自适应地聚集足够的搭配点，以实现快速的光谱收敛。我们的缩放算法在频域中使用一个指标，该指标既用于确定何时需要缩放，又用于通知缩放因子的调整以重新分配搭配点以适应解决方案的扩散行为。我们的移动技术采用外部误差指示器并移动搭配点以捕获翻译。频率和外部误差指标均仅使用数值解来定义。我们将我们的方法应用于许多不同的模型，包括扩散和移动的费米-狄拉克分布和非线性狄拉克孤立波，并证明了瞬态模拟的光谱收敛恢复。解决线性抛物线问题的性能比较表明，我们的频率缩放算法优于现有的缩放方法。我们还展示了我们的频率缩放技术能够跟踪细胞增殖模型中平均细胞大小的膨胀。除了具有无穷远指数衰减的 Laguerre 和 Hermite 基函数之外，