Jacobian-free Newton–Krylov methods with GPU acceleration for computing nonlinear ship wave patterns

• The paper is concerned with computing steady three-dimensional ship wave patterns.
• As a case study, we consider free-surface flows past a submerged source.
• A singular integro-differential equation is solved via collocation and Newton's method.
• We apply Jacobian-free Newton–Krylov methods with a carefully chosen banded preconditioner.
• Further grid refinement and decreased run-time is obtained by GPU acceleration.

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.

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