谱元方法求解波动方程及影响其数值精度的相关因素

Implicit Spectral Element Method for Wave Equation and Some Factors Influencing Numerical Accuracy

为探讨波动方程的高精度数值模拟,采用Chebyshev谱元方法结合隐式Newmark时间积分方法求解波动方程.求解一个具体算例验证了数值方法的可行性,讨论了时间步长、Newmark因子以及计算区域的网格剖分方式对数值精度的影响.结果表明:和差分法相比,谱元方法求解波动方程具有所用网格节点少,数值精度高的特点;数值误差随时间步长减小而减小;在满足稳定性要求的前提下,数值误差随着Newmark因子的减小而减小;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.所述方法和结论可用于模拟声波在空气中的传播.

To investigate the numerical scheme with high order of accuracy for the simulation of wave equations, Chebyshev spectral element method combined with implicit Newmark time integral method is adopted for simulating wave equations. Then some factors affecting the numerical accuracy are discussed in detail, such as time step h, Newmark factor and the subdivision style for computational domain. The conclusions indicate that the spectral element method has higher order numerical accuracy than difference method for simulation of wave equations, small time step induces less numerical error, smaller Newmark factor induces smaller numerical error, and different mesh style induces different numerical error while the general grid member gets equal to each other. The proposed method enables to simulate acoustic wave propagation in air.