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

Abstract

We presented a novel approach to investigate the two-dimensional shallow water equation in its primitive form. Its employs the P1NC−P1 element pair to simulate various cases: standing waves, dam-break planar, and wave absorbing with embedded radiation boundary conditions. Unlike the conventional method, we approximate the free surface variable using a conformal basis P1 whereas the velocity potential is approximated using a non-conformal basis, P1NC. Thus, for each case, the weak form needs to be reformulated as well as the discrete form. The resulting scheme is a first-order ordinary differential system and solved by Crank Nicholson. The mass matrix in the momentum equation contains the multiplication between the two bases, which computed by the mass lumping. So, our method is explicit, flexible and easy to implement. Validation using standing waves demonstrated first-order accuracy, free from numerical damping and convergent to the analytical solution. Dam-break simulation result shown an agreement with ANUGA software. Our scheme's flexibility is demonstrated when it can mimic wave absorbing simulation employing embedded radiation boundary conditions. The reflection at the boundary seems small enough, thus can be neglected. All these findings have shown the robustness and capability of our scheme to predict accurate results for various shallow water flow problems. • A novel technique for solving 2D SWE in primitive form • It is explicit, flexible, easy to implement, accurate, and robust • Our approach is suitable for coastal/oceanographic simulations

Keywords

• The P1NC−P1 Finite Element Method