Spatial and temporal discretization of surface flow equations: Implications for modeling stability and accuracy
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Partial differential equations used to describe surface water flow can only be solved analytically for limited sets of initial and boundary conditions rarely occurring in nature. The use of numerical methods such as finite elements or finite differences is common to approximate the solution of surface water flow equations for naturally occurring complex conditions. The accuracy and stability of the numerical solutions depend on the numerical scheme, the time step and the grid size. While forward and backward schemes stability is dependent on the Courant Number, central schemes are unconditionally stable. Nevertheless these central schemes (e.g., Galerkin finite element) tend to generate spurious oscillations that affect the accuracy of the solution. These oscillations have been attributed to phase errors in the Fourier components of the solution. An innovative method is developed to generate dynamic time step criteria that ensure the accuracy of the solution of such problems. These dynamic time step criteria are generated using a series of numerical experiments combined with a regression analysis. The dynamic time step criteria are normalized by a problem specific time factor to generalize the equation. Dynamic time step criteria are developed and tested for 1-D and 2-D kinematic wave equations using the consistent and the lumped Galerkin finite elements formulations, respectively and for the 1-D full St. Venant equations using the Abbot-Ionesco finite differences formulation.
