Stability and accuracy of finite element schemes for the one-dimensional kinematic wave solution

Solving the kinematic wave equations for overland flow using the conventional consistent Galerkin finite element scheme is known to result in numerical oscillations due to the non-symmetric first spatial derivative terms in the kinematic wave equations. In this paper the lumped and the upwind finite element schemes are evaluated as alternatives to the consistent Galerkin finite element scheme. Stability analysis of the upwind scheme shows that the damping effect, that could reduce the oscillations, is small for the high Courant numbers encountered in overland flow problems. The upwind scheme, using upwind factors of 0.1 and 1.0, did not provide any improvement to the stability of the lumped and the consistent schemes. The lumped scheme considerably reduces oscillations without significant reduction in the overall solution accuracy. No analytical guidelines for time-step criteria that will insure stability and accuracy were provided by the stability analysis performed for the three schemes. Problem specific accuracy-based dynamic time-step criteria was developed and evaluated for the lumped scheme. These time-steps were found to be on average, double the size of the dynamic time-steps for the consistent scheme.