Implicit Free Surface: Velocity Formulation

Equation () can be solved explicitly for the depth field at the arrival point in terms of the divergence of the velocity field as

where

Further, () can be simplified to express in terms of as follows,

where

If we then insert () into () and assume as known, then we have a linear elliptic equation for ,

where the right-hand-side is known and is the linear elliptic operator such that

Equations () and () constitute our implicit free surface semi-Lagrangian shallow water model formulated in terms of an elliptic equation for the velocity field.