非线性发展方程的时空高精度谱配置算法

谱方法是求解偏微分方程的一种重要数值方法，已被广泛应用于科学及工程计算的众多领域。传统的计算方法在时间方向主要采用差分方法，导致与空间方向的谱逼近不匹配，从而限制了整体精度的提高。因此，本项目拟研究非线性发展方程的时空高精度谱配置算法。我们首先考虑非线性常微分方程(组)初值问题的Legendre配置方法，并提出简单易施、快速稳定的高精度算法；然后考虑非线性抛物型和双曲型方程初边值问题，采用Legendre配置法对空间和时间方向进行全离散，进而构造相应的时空高精度算法。本项目的研究成果将进一步推动谱方法的应用，并为实际计算中更为复杂的发展型偏微分方程的数值求解提供强有力的算法支持。

Spectral method is an important numerical method for solving partial differential equations which has been widely used in many areas of scientific and engineering computing. The difference scheme is usually adopted in time, which resulted in the mismatch with the spectral scheme in the spatial direction, and limited the overall accuracy. Thus, in this project we shall focuse on study of the high accuracy algorithm for the space-time spectral collocation method for nonlinear evolution equations. We first consider the initial value problems of nonlinear ordinary differential equations, and design simple,fast and stable Legendre collocation algorithms with high accuracy. We then consider the initial boundary value problems of nonlinear parabolic and hyperbolic equations, use the Legendre collocation method for discretization in both spatial and temporal directions, and design the corresponding space-time algorithms with high accuracy. The research results of this project will further promote the application of spectral method, and provide a powerful support for numerically solving the more complicated evolutionary partial differential equations in the actual computation.