Selective application of a total variation diminishing term to an implicit method for two-dimensional flow modelling

An implicit method with selective total variation diminishing (TVD) term inclusion was developed and used to find solutions to the shallow water equations (SWEs) for free-surface flows in natural and engineered channels. The conservative form of the SWEs was employed, and the method incorporated an algorithm for selectively underrelaxing the iterative process to maintain stability and accuracy in the presence of shock interfaces. The value of the Courant number and the frequency at which the TVD term was incorporated were constantly updated during the computation to achieve optimal speed of execution while maintaining stability. The method was tested against published results from physical experiments and from computations employing alternative algorithms, and the results obtained demonstrate both the economy and accuracy of the proposed algorithm