Galerkin时空耦合谱元法求解声波动方程

提出了时空耦合谱元方法,并将其用于带第一类边界条件的非齐次一维、二维、三维波动方程的求解。分别采用四边形、六面体和超六面体作为计算单元,在每个单元内采用Chebyshev多项式的极值点作为Lagrange插值节点,并且探讨了区域剖分方式对计算精度的影响。时空耦合谱元法能够得到精度很高的数值结果,并且其色散随时间推移是稳定的;当总网格节点数相同时,不同的网格剖分方式所得数值误差不同,当空间方向Chebyshev多项式的阶数较高和时间方向Chebyshev多项式的阶数较低时,得到的数值精度较高;在总节点数相同的情况下,与时间全域方式相比,逐时间子区域方式计算所需要的时间更经济,两种方式可以得到相同的精度。结果表明:时空耦合谱元方法使时空方向精度相匹配,可以提高整体精度;空间方向的Chebyshev多项式对数值精度起主要影响作用;时间子区域方式的采用可以扩大问题的计算区域。 

The research of space-time coupled spectral element method for acoustic wave equation

A space-time coupled spectral element method based on Chebyshev polynomials is presented for solving timedependent wave equations.Acoustic propagation problems in 1 + 1,2 + 1,3 + 1 dimensions with the dirichlet boundary conditions are simulated via space-time spectral element method using quadrilateral,hexahedral and tesseractic elements respectively.Space-time coupled spectral element method can obtain high-order precision and the dispersion is stable over time.With the same total number of grid nodes,higher numerical precision is obtained if the higher-order Chebyshev polynomials in space directions and lower-order Chebyshev polynomials in time direction are adopted.When space-time coupled spectral element method is used,time subdomain-by-subdomain approach is more economical than time domain approach.