An implicit time‐marching algorithm for shallow water models based on the generalized wave continuity equation

Abstract

Wave equation models currently discretize the generalized wave continuity equation with a three-time-level scheme centered at k and the momentum equation with a two-time-level scheme centered at k+1/2; non-linear terms are evaluated explicitly. However in highly non-linear applications, the algorithm becomes unstable at even moderate Courant numbers. This paper examines an implicit treatment of the non-linear terms using an iterative time-marching algorithm. Depending on the domain, results from one-dimensional experiments show up to a tenfold increase in stability and temporal accuracy. The sensitivity of stability to variations in the G-parameter (a numerical weighting parameter in the generalized wave continuity equation) was examined; results show that the greatest increase in stability occurs with G/τ=2–50. In the one-dimensional experiments, three different types of node spacing techniques—constant, variable, and LTEA (Localized Truncation Error Analysis)—were examined; stability is positively correlated to the uniformity of the node spacing. Lastly, a scaling analysis demonstrates that the magnitudes of the non-linear terms are positively correlated to those that most influence stability, particularly the term containing the G-parameter. It is evident that the new algorithm improves stability and temporal accuracy in a cost-effective manner. Copyright © 2001 John Wiley & Sons, Ltd.