Strongly imposing the free surface boundary condition for wave equations with finite difference operators

Longfei Gao
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引用次数: 1

Abstract

Acoustic and elastic wave equations are routinely used in geophysical and engineering studies to simulate the propagation of waves, with a broad range of applications, including seismology, near surface characterization, non-destructive structural evaluation, etc. Finite difference methods remain popular choices for these simulations due to their simplicity and efficiency. In particular, the family of finite difference methods based on the summation-by-parts operators and the simultaneous-approximation-terms technique have been proposed for these simulations, which offers great flexibility in addressing boundary and interface conditions. For the applications mentioned above, surface of the earth is usually associated with the free surface boundary condition. In this study, we demonstrate that the weakly imposed free surface boundary condition through the simultaneous-approximation-terms technique can have issue when the source terms, which introduces abrupt disturbances to the wave field, are placed too close to the surface. In response, we propose to build the free surface boundary condition into the summation-by-parts finite difference operators and hence strongly and automatically impose the free surface boundary condition to address this issue. The procedure is very simple for acoustic wave equation, requiring resetting a few rows and columns in the existing difference operators only. For the elastic wave equation, the procedure is more involved and requires special design of the grid layout and summation-by-parts operators that satisfy additional requirements, as revealed by the discrete energy analysis. In both cases, the energy conserving property is preserved. Numerical examples are presented to demonstrate the effectiveness of the proposed approach.
对具有有限差分算子的波动方程强施加自由表面边界条件
声波和弹性波动方程通常用于地球物理和工程研究中,以模拟波的传播,具有广泛的应用范围,包括地震学,近地表表征,非破坏性结构评价等。有限差分法由于其简单和高效,一直是这些模拟的流行选择。特别地,基于分部求和算子和同时逼近项技术的有限差分方法族已经被提出用于这些模拟,它在处理边界和界面条件方面提供了很大的灵活性。对于上述应用,地球表面通常与自由表面边界条件联系在一起。在这项研究中,我们证明了通过同时逼近项技术弱施加的自由表面边界条件在源项太靠近表面时可能会出现问题,这会给波场带来突然的干扰。因此,我们建议将自由曲面边界条件构建为分部求和有限差分算子,从而强而自动地施加自由曲面边界条件来解决这一问题。该方法对于声波方程非常简单,只需要在已有的差分算子中重新设置几行和几列即可。对于弹性波动方程,离散能量分析表明,这一过程更为复杂,需要对满足附加要求的网格布局和分部求和算子进行特殊设计。在这两种情况下,能量守恒。数值算例验证了该方法的有效性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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