Purpose

The purpose of this study is twofold: first, to derive a consistent model free-surface runup flow problems over deformable beds. The authors couple the nonlinear one-dimensional shallow water equations, including friction terms for the water free-surface and the two-dimensional second-order solid elastostatic equations for the bed deformation. Second, to develop a robust hybrid finite element/finite volume method for solving free-surface runup flow problems over deformable beds. The authors combine the finite volume for free-surface flows and the finite element method for bed elasticity.

Design/methodology/approach

The authors propose a new model for wave runup by static deformation on seabeds. The model consists of the depth-averaged shallow water system for the water free-surface coupled to the second-order elastostatic formulation for the bed deformation. At the interface between the water flow and the seabed, transfer conditions are implemented. Here, hydrostatic pressure and friction forces are considered for the elastostatic equations, whereas bathymetric forces are accounted for in the shallow water equations. As numerical solvers, the authors propose a well-balanced finite volume method for the flow system and a stabilized finite element method for elastostatics.

Findings

The developed coupled depth-averaged shallow water system and second-order solid elastostatic system is well suited for modeling wave runup by deformation on seabeds. The derived coupling conditions at the interface between the water flow and the bed topography resolve well the condition transfer between the two systems. The proposed hybrid finite volume element method is accurate and efficient for this class of models. The novel technique used for wet/dry treatment accurately captures the moving fronts in the computational domain without generating nonphysical oscillations. The presented numerical results demonstrate the high performance of the proposed methods.

Originality/value

Enhancing modeling and computations for wave runup problems is at an early stage in the literature, and it is a new and exciting area of research. To the best of our knowledge, solving wave runup problems by static deformation on seabeds using a hybrid finite volume element method is presented for the first time. The results of this research study, and the research methodologies, will have an important influence on a range of other scientists carrying out research in related fields.

中文翻译：

---

基于海床静态变形的浅水波有限体积/有限元混合方法

目的

这项研究的目的有两个：首先，在可变形床上推导出一致的模型自由表面爬升流动问题。作者耦合了非线性一维浅水方程，包括水自由表面的摩擦项和床变形的二维二阶固体弹性静力学方程。其次，开发一种稳健的混合有限元/有限体积方法，用于解决可变形床上的自由表面爬升流动问题。作者结合了自由表面流动的有限体积和床弹性的有限元方法。

设计/方法/方法

作者提出了一种新的海床静态变形引起的波浪爬高模型。该模型由水自由表面的深度平均浅水系统与床变形的二阶弹性静力学公式耦合组成。在水流和海床之间的界面处，实施了转移条件。在这里，弹性静力学方程考虑了静水压力和摩擦力，而浅水方程考虑了测深力。作为数值求解器，作者提出了一种用于流动系统的均衡有限体积方法和一种用于弹性静力学的稳定有限元方法。

发现

开发的耦合深度平均浅水系统和二阶固体弹性静力学系统非常适合模拟海底变形引起的波浪爬高。在水流和床地形之间的界面处导出的耦合条件很好地解决了两个系统之间的条件转移。所提出的混合有限体积元方法对于此类模型是准确且有效的。用于湿/干处理的新技术准确地捕获了计算域中的移动前沿，而不会产生非物理振荡。所提出的数值结果证明了所提出方法的高性能。

原创性/价值

增强波浪爬高问题的建模和计算处于文献的早期阶段，这是一个新的令人兴奋的研究领域。据我们所知，首次提出了使用混合有限体积元方法通过海床静态变形解决波浪爬高问题。这项研究的结果和研究方法将对在相关领域进行研究的其他一系列科学家产生重要影响。