This article focuses on the design of semi-implicit schemes that are fully well-balanced
for the one-dimensional shallow water equations, that is, schemes that preserve all smooth steady
states of the system and not just water-at-rest equilibria. The proposed methods outperform standard
explicit schemes in the low-Froude regime, where the celerity is much larger than the fluid velocity,
eliminating the need for a large number of iterations on large time intervals. In this work, splitting
and relaxation techniques are combined in order to obtain fully well-balanced semi-implicit first
and second order schemes. In contrast to recent Lagrangian-based approaches, this one allows the
preservation of all the steady states while avoiding the complexities associated with Lagrangian
formalism.
