  The nonlinear problem of steady free-surface flow past a submerged source is
considered as a case study for three-dimensional ship wave problems. Of
particular interest is the distinctive wedge-shaped wave pattern that forms on
the surface of the fluid. By reformulating the governing equations with a
standard boundary-integral method, we derive a system of nonlinear algebraic
equations that enforce a singular integro-differential equation at each
midpoint on a two-dimensional mesh. Our contribution is to solve the system of
equations with a Jacobian-free Newton-Krylov method together with a banded
preconditioner that is carefully constructed with entries taken from the
Jacobian of the linearised problem. Further, we are able to utilise graphics
processing unit acceleration to significantly increase the grid refinement and
decrease the run-time of our solutions in comparison to schemes that are
presently employed in the literature. Our approach provides opportunities to
explore the nonlinear features of three-dimensional ship wave patterns, such as
the shape of steep waves close to their limiting configuration, in a manner
that has been possible in the two-dimensional analogue for some time.
