This work is motivated by two observations related to numerical solutions for nonlinear wave-structure interaction. First, the combination of an Immersed Boundary Method (IBM) for the body-boundary and a σ transform for the free-surface and bottom boundary, as adopted by [1], requires the construction of an artificial C2 continuous freesurface inside the body. While this can be done with some success in 2D, it is likely to be much more problematic in 3D. Secondly, the work of [2] shows quite promising results for the application of an IBM technique for all domain boundaries in the context of the Harmonic Polynomial Cell (HPC) method. Thus, the goal of the present work is to investigate the accuracy and efficiency of a finite difference method solution where all of the fluid domain boundaries are introduced using the IBM. We establish the accuracy and convergence of the method for nonlinear wave propagation on both constantand variable-depth fluids, demonstrating that the accuracy is comparable to that achieved by the σ transform approach. A preconditioned GMRES iterative solution strategy is also developed and shown to give optimal scaling of the solution effort in 2D.

中文翻译：

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使用浸入边界法求解非线性水波的有限差分解

这项工作的动机是与非线性波结构相互作用的数值解相关的两个观察结果。首先，[1] 采用的身体边界的浸入边界方法 (IBM) 和自由表面和底部边界的 σ 变换的组合需要在身体内部构建人工 C2 连续自由表面。虽然这可以在 2D 中取得一些成功，但在 3D 中可能会出现更多问题。其次，[2] 的工作显示了在谐波多项式单元 (HPC) 方法的上下文中将 IBM 技术应用于所有域边界的非常有希望的结果。因此，当前工作的目标是研究有限差分方法解决方案的准确性和效率，其中使用 IBM 引入所有流体域边界。我们建立了在恒定和可变深度流体上非线性波传播方法的精度和收敛性，证明精度与 σ 变换方法所达到的精度相当。还开发并展示了预处理的 GMRES 迭代解决方案策略，以在 2D 中优化解决方案工作量。