The Pi − P1NC Finite Element Method for 1D wave simulation using Shallow Water Equations

Abstract

We study a simple numerical scheme based on a new type of Finite Element Method (FEM) to solve the 1D Shallow Water Equations. In the new scheme, the surface elevation variable is approximated by a linear continuous basis function (P 1) and the velocity potential variable is approximated by the one-dimensional discontinuous linear non-conforming basis function ( P 1 N C ). Here, we implement the P 1 − P 1 N C finite element pair to solve the 1D Shallow Water Equations on a structured grid, whereas the Runge Kutta method is adopted for time integration. We verified the resulting scheme by conducting several simulations such as a standing wave simulation, and propagation of an initial hump over sloping bathymetry. The resulting scheme free from numerical damping error, conservative and both standing wave and shoaling phenomena are well simulated.

A novel technique for implementing the finite element method in a shallow water equation

“…In this paper, we propose a novel technique to solve SWE using finite element method (FEM) based on our previous research [ 28 ]. In our prior work, we proposed a novel one-dimensional basis function, the discontinuous (non-conformal) .…”