We construct A-stable and L-stable diagonally implicit Runge-Kutta methods of which the diagonal vector in the Butcher matrix has a minimal maximum norm. If the implicit Runge-Kutta relations are iteratively solved by means of the approximately factorized Newton process, then such iterated Runge-Kutta methods are suitable methods for integrating shallow water problems in the sense that the stability boundary is relatively large and that the usually quite fine vertical resolution of the discretized spatial domain is not involved in the stability condition.

Multistep, Runge-Kutta and extrapolation methods (msc 65L06), Stability and convergence of numerical methods (msc 65L20), Stability and convergence of numerical methods (msc 65M12), Method of lines (msc 65M20)
CWI
Modelling, Analysis and Simulation [MAS]
Computational Dynamics

van der Houwen, P.J, & Sommeijer, B.P. (1999). Diagonally implicit Runge-Kutta methods for 3D shallow water applications. Modelling, Analysis and Simulation [MAS]. CWI.