Difference schemes with weights for modelling fluid flows in the shallow water approximation

Mathematical modelling of free boundary fluid flows is often based on the shallow water approximation. The system of equations includes a scalar advection equation for fluid height and a vector advection equation for velocity. The paper presents an approximation of the initial boundary value problem using the standard spatial finite element method. Time weights are used in implicit two-level schemes. The computational method is based on Newton's method. The fulfillment of the laws of conservation of mass and total mechanical energy at the continuous and discrete level is discussed. Numerical results demonstrate the effectiveness of implicit schemes for approximating solutions to one- and two-dimensional model problems, specifically dam failure. It is shown that increasing the weight in the two-level scheme improves the monotonicity of the approximate solution.