Time Discretization Schemes for Poincaré Waves in Finite-Element Shallow-Water Models

The finite-element spatial discretization of the linear shallow-water equations is examined in the context of several temporal discretization schemes. Three finite-element pairs are considered, namely, the $P^{}_{0}-P^{}_{1}$, $P^{NC}_{1}-P^{}_{1}$, and $RT^{}_{0}-P^{}_{0}$ schemes, and the backward and forward Euler, Crank-Nicolson, and second and third order Adams-Bashforth time stepping schemes are employed. A Fourier analysis is performed at the discrete level for the Poincaré waves, and it determines the stability limit of the schemes and the error in wave amplitude and phase that can be expected. Numerical solutions of test problems to simulate Poincaré waves illustrate the analytical results.