声波方程高阶有限差分数值模拟中的平面波透射边界条件

Application of plane-wave transmitting boundary conditions in high-orderfinite-difference numerical simulation using acoustic wave equations

一种简单高效的吸收边界对于提高大规模波动方程数值模拟的计算效率十分关键。基于平面波传播理论的Liao氏透射边界(Liao边界)具有编程简单、耗费内存少且计算效率高的优点,三阶Liao边界对大角度入射波的吸收效果较好,但存在不稳定的问题。为此,从理论上分析了三阶Liao边界的稳定性,并提出一种基于修改插值权重的稳定化策略,避免了三阶Liao边界长时间模拟的不稳定问题,同时并未显著降低其吸收效果。此外,分别针对二阶和一阶声波方程,详细讨论了Liao边界在中心和交错网格高阶有限差分模拟中的实现方法。三维波场数值模拟算例表明,结合混合吸收边界思路的二阶和三阶Liao边界的吸收效果明显优于常规分裂式完美匹配层边界,并且三阶Liao边界的吸收效果接近复频移卷积型完美匹配层边界。在计算效率方面,当采用中心网格有限差分求解二阶声波方程时,使用10层Liao边界所增加的计算量接近10%,远低于常规分裂式完美匹配层边界。将其用于求解一阶声波方程,Liao边界的计算成本也明显低于使用复频移卷积型完美匹配层边界。

A simple efficient absorbing boundary condition(ABC)is important for large-scale wave equation numerical simulation.The Liao s transmitting boundary condition(Liao boundary for short)based on plane-wave propagation has the advantages of simple programming,low memory consumption,and high computational efficiency.The third-order Liao boundary performs well for large-angle incident waves.However,it suffers from numerical instability.We analyze the reason for instability and propose a stabilization strategy for long-time simulation by modifying an interpolation parameter in the third-order Liao ABC.We also discuss in detail the implementation of Liao ABC for the second-and first-order acoustic wave equations,which are usually solved by using high-order centered-and staggered-grid finite-difference(FD)methods,respectively.We combine the idea of hybrid ABC and the Liao formula to improve absorption performance further.Several three-dimensional simulation examples show that the second-and third-order hybrid Liao ABCs can absorb waves much cleaner than the conventional split perfectly matched layer(SPML),and the absorption effect of the third-order Liao boundary is close to that of the complex-frequency shift convolutional PML(CFS-CPML).In terms of computational efficiency,the application of a ten-layer Liao ABC to the second-order acoustic wave equation using the centered-grid FD method increases the computing time approximately by 10%,which is much lower than of SPML.In comparison with CFS-CPML,the application of a Liao ABC to the first-order acoustic wave equation also saves computing time visibly.