多边形区域上高波数Helmholtz问题的新型数值Fokas解法研究

Fokas方法（或统一变换）是求解Helmholtz问题的常用方法，也是突破高波数问题的重要途径。本项目结合多频振荡数值积分算法与奇异振荡积分方程算法，研究多边形区域上高波数Helmholtz问题的数值Fokas方法，并构造实现Fokas方法的快速、高精度算法。具体研究内容包括：1）针对Fokas方法中的多频振荡曲线积分，设计新型Levin方法；2）针对一般边值问题，利用Riemann-Hilbert问题、Filon方法，设计适用范围更广的高性能算法；3）基于振荡积分理论研究数值解关于波数的渐近性分析。本项目的预期研究成果既将促进高波数Helmholtz问题数值计算以及数值Fokas方法的发展，也将丰富可积系统、随机矩阵等领域中Riemann-Hilbert问题数值解的研究内容。

As a frequently-used method for solving Helmholtz problems, the Fokas method, or known as the unified transform, is an important approach to beat high-wavenumber problems. By combining quadrature rules of oscillatory multifrequency integrals and numerical solutions of singular and oscillatory integral equations, this project is restricted to studying the numerical Fokas method for solving Helmholtz equations in polygons with high wavenumbers, and developing a fast and high-order algorithm for implementing the Fokas method. The specific contents of this project are listed as follows: 1) new Levin methods are constructed to computing oscillatory multifrequency curvilinear integrals; 2) high-performance algorithms for general boundary value problems are devised with the help of the Riemann-Hilbert problem and the Filon method; 3) asymptotic properties of numerical solutions with respect to wavenumbers are studied based on the oscillatory integral theory. Expected results of this project will not only promote the progresses in both numerical solutions of Helmholtz problems with high wavenumbers and the numerical Fokas method, but also enrich numerical studies on Riemann-Hilbert problems arising in the integrable system, the random matrix theory and etc.