Stability and accuracy of the finite element solution to the kinematic wave overland flow

The numerical solution of the kinematic wave equations for overland flow is known for its numerical oscillations. The conventional Galerkin finite element consistent formulation results in violation of physical reality and numerical oscillations. Another cause of oscillations is the first derivative terms in the kinematic wave equations that result in non-symmetric operators that are very oscillatory in nature. The lumped formulation is tested in this paper as an alternative to the consistent formulation. It was found that the lumped formulation significantly improves stability of the solution with no reduction in accuracy. The upwind finite element method is also tested to remedy the oscillations caused by the first derivative. The latter scheme is applied to both, lumped and consistent formulations, using upwind factors of 0.1 and 1.0 and is evaluated in terms of stability and accuracy of the solution. The upwind method did not provide any improvement to the stability of both the lumped and the consistent formulation. It required smaller or equal time steps compared to a non-upwinded formulation for the same level of accuracy. The lumped formulation is recommended to solve overland flow problems as it was found to be the most efficient formulation to solve the 1-D kinematic wave equation for overland flow. The dynamic time step for the lumped formulation developed in this study can be easily integrated in overland flow routing models to guide the choice of the optimal time step with minimum user input.