We propose a high-order curvilinear Lagrangian finite element method for shallow water hydrodynamics. This method falls into the high-order Lagrangian framework using curvilinear finite elements. We discretize the position and velocity in continuous finite element spaces. The high-order finite element basis functions are defined on curvilinear meshes and can be obtained through a high-order parametric mapping from a reference element. Considering the variational formulation of momentum conservation, the global mass matrix is independent of time due to the use of moving finite element basis functions. The mass conservation is discretized in a pointwise manner which is referred to as strong mass conservation. A tensor artificial viscosity is introduced to deal with shocks, meanwhile preserving the symmetry property of solutions for symmetric flows. The generic explicit Runge–Kutta method could be adopted to achieve high-order time integration. Several numerical experiments verify the high-order accuracy and demonstrate good performances of using high-order curvilinear elements.

中文翻译：

---

浅水流体动力学的高阶曲线拉格朗日有限元方法

我们提出了一种用于浅水流体动力学的高阶曲线拉格朗日有限元方法。该方法属于使用曲线有限元的高阶拉格朗日框架。我们离散连续有限元空间中的位置和速度。高阶有限元基函数在曲线网格上定义，并且可以通过参考元素的高阶参数映射来获得。考虑动量守恒的变分公式，由于使用移动有限元基函数，全局质量矩阵与时间无关。质量守恒以点方式离散化，称为强质量守恒。引入张量人工粘度来处理冲击，同时保留对称流解的对称性。可以采用通用的显式龙格-库塔方法来实现高阶时间积分。多个数值实验验证了高阶精度并展示了使用高阶曲线单元的良好性能。