基于坐标变换求解奇异问题的高精度谱配置方法研究

谱方法是科学计算中的一类重要工具，它最显著的优点是谱精度。但在实际中许多函数和微分方程的解具有端点或拐点奇异性。此时利用传统的谱方法求解，其优越性就不再凸显，因为函数的正则性越差，传统谱方法的谱精度就越低。近年来利用谱方法求解边界奇异问题取得了很大的进展，且被成功地应用到很多实际问题的求解中。但是现有方法大都没法得到指数收敛的解。为恢复谱方法的指数收敛性，拟引入可削减解奇性的坐标变换，进而在新的坐标下设计快速收敛的广义有理Chebyshev配置点谱方法和Fourier区域截断法，并分析其收敛性和误差。同时，将给出的算法应用到奇异函数逼近、奇异数值积分和奇异微分方程求解中。本项目的优点为：其一、对奇异问题可以快速得到其收敛的谱级数解。其二、不同于一般的有理谱方法，其实用性更强且求解微分矩阵计算量更小、误差更易分析、程序更易实现。该工作将进一步拓展有理谱方法的应用范围，丰富微分方程的数值解。

Spectral methods are powerful numerical tools in scientific computing. The most fascinating merit of spectral methods is spectral accuracy. However, in fact, many functions and solutions of differential equations own the singularities, especially for the non-smooth boundary problems. The singularity occurs more evidently at the boundaries or corners. It is a pity that the most outstanding advantage of spectral methods will disappear. Recently, great progress has been made in solving singularity problems by spectral methods. However, most of them can not obtain spectral accuracy. To recover the exponential accuracy of spectral methods, general rational Chebyshev collocation spectral methods and truncated Fourier spectral methods with coordinate transformations are introduced. Then we use spectral methods to solve the problems in the new coordinates. Furthermore, we will analyse the convergence and give error estimates. At the same time, we will use the proposed methods to approximate the singularity functions and to solve the singularity quadrature and singularity differential equations. There are two aspects to make it stand out from others. One of them is one can obtain high accuracy solution for singularity problems. The other aspect is to reduce the complexity of computing differential matrices and easier to analyse errors and easier to implement compared to the traditional rational spectral methods. Overall, the program will extend the applications of rational spectral methods and enrich the methods for solving differential equations.