High-order curvilinear Lagrangian finite element methods for shallow water hydrodynamics

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.