Recently, new adaptive techniques were developed that greatly improved the efficiency of solving PDEs using spectral methods. These adaptive spectral techniques are especially suited for accurately solving problems in unbounded domains and require the monitoring and dynamic adjustment of three key tunable parameters: the scaling factor, the displacement of the basis functions, and the spectral expansion order. There have been few analyses of numerical methods for unbounded domain problems. Specifically, there is no analysis of adaptive spectral methods to provide insight into how to increase efficiency and accuracy through dynamical adjustment of parameters. In this paper, we perform the first numerical analysis of the adaptive spectral method using generalized Hermite functions in both one- and multi-dimensional problems. Our analysis reveals why adaptive spectral methods work well when a "frequency indicator" of the numerical solution is controlled. We then investigate how the implementation of the adaptive spectral methods affects numerical results, thereby providing guidelines for the proper tuning of parameters. Finally, we further improve performance by extending the adaptive methods to allow bidirectional basis function translation, and the prospect of carrying out similar numerical analysis to solving PDEs arising from realistic difficult-to-solve unbounded models with adaptive spectral methods is also briefly discussed.

翻译：最近,开发了新的适应技术,大大提高了利用光谱方法解决PDE的效率,这些适应性光谱技术特别适合于准确解决无限制领域的问题,需要监测和动态调整三种关键的可金枪鱼参数:缩放系数、基函数的偏移和光谱扩展顺序。对无限制域问题的数字方法分析很少。具体地说,没有分析适应性光谱方法,以深入了解如何通过动态调整参数来提高效率和准确性。在本文中,我们用一维和多维问题中的通用赫米特函数对适应性光谱方法进行第一次数字分析。我们的分析揭示了为什么在控制数字解决方案的“频率指标”时,适应性光谱方法效果良好。然后我们调查适应性光谱方法的实施如何影响数字结果,从而为适当调整参数提供了指导方针。最后,我们通过扩展适应性方法,允许双向基础函数转换,从而进一步改进性能。我们利用现实的难以解析的光谱模型进行类似的数字分析的前景也是与适应性光谱方法一起讨论的解决PDE。