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

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