五阶偏微分方程的时间多区域时空谱方法

Multi-Domain Space-Time Spectral Method in Time for Solving Fifth-Order Partial Differential Equation

摘要: 本文针对五阶偏微分方程提出了时间多区域时空谱方法. 该方法在空间方向上采用 Legendre-Petrov-Galerkin 方法, 在时间方向上采用多区域的 Legendre-tau 方法, 即: 把时间区间分成多个区域, 并在每个区域上采用 Legendre-tau 方法. 同时, 本文给出了该方法在线性问题上的误差分析, 选取适当的基函数使得系数矩阵稀疏, 对非线性方程中的非线性项采用在 Chebyshev-Gauss- Lobatto 点上的插值进行计算. 最后通过一些数值算例验证了算法的有效性.

Abstract: In this paper, we propose the multi-domain space-time spectral method in time for the fifth-order partial differential equation. Legendre-Petrov-Galerkin method is applied in space direction, and multi-domain Legendre-tau method is applied in time direction, that is: dividing the time interval into multiple domains and Legendre-tau method is applied in each domain. At the same time, the error analysis of the method on linear problems is given in this paper, and we choose proper basis functions to make the coefficient matrix sparse, the nonlinear term for the nonlinear equation is computed by interpolation through Chebyshev-Gauss-Lobatto points. Finally, some numerical examples are given to verify the effectiveness of the algorithm.