基于高维椭球波函数的谱方法及其应用

In recent years, the spectral methods using prolate spheroidal wave functions (PSWFs) and their applications for high frequency problems become a hot research topic in the area of scientific computing. The prolate spheroidal wave functions are a family of orthogonal bandlimited functions which have quasi-uniformly distributed zeros, and enable fewer points per wavelength to resolve waves. They appear superior to polynomials in simulating high frequency problems. With these advantageous features, the PSWFs-based spectral methods have great significance for achieving high accuracy, keeping stabilization of numerical result and solving high frequency problems in complex geometries. However, the theory and algorithm of PSWFs-based spectral methods still need further development, enrichment and complement, from single domain to complex domain..In this project, our endeavor is devoted to the development of PSWFs-based spectral method. More specifically, 1) we will propose and analyze high-dimensional PSWFs-based spectral method；2) we will study the efficient PSWFs-based spectral solver; 3) we will develop PSWFs-based spectral multi-domain method for high wave number problems.

基于椭球波函数的谱方法及其在高频问题中的应用是近年来科学计算的一个研究热点。椭球波函数是一类具有双重正交性的有限带宽函数，它同时是一类微分算子与积分算子的特征值函数，具有拟一致分布的零点。与多项式相比，它对高频问题具有更佳的逼近性质，单位波长所需的模数更少，从而可以以非常小的自由度取得指定精度，达到问题求解的高效自适应性。 因此，研究基于高维椭球波函数的各类谱方法及其应用，对于拓展谱方法的基础理论，发展和丰富微分方程的数值解法有着重要的科学意义。 . 基于此，本项目将研究如下内容：（1）研究基于高维椭球波函数的谱方法的基础算法和理论；（2）设计基于椭球波函数的快速算法；（3）椭球波函数谱方法在大波数复杂散射体问题中的应用。