Numerical solutions to Zermelo’s navigation problem under variable ocean current fields

Zermelo’s navigation problem seeks the time-optimal heading strategy for a vessel of fixed speed navigating through a variable ocean current field. This paper reformulates the problem as an optimal control problem governed by the dynamic Hamilton-Jacobi-Bellman (HJB) equation and implements two complementary numerical solvers: a grid-based Eulerian level-set scheme (Method I) and a Lagrangian extremal field algorithm (Method II). Convergence is analysed within the viscosity solution framework, yielding (ℎ1/2) and (ℎ) rate bounds under Lipschitz and semiconcavity conditions, respectively. Grid refinement studies on a Rankine vortex and a double-gyre circulation confirm that Method I achieves 𝑂(ℎ3/2) in smooth regimes and degrades to 𝑂(ℎ0.48) under strong currents, while Method II reaches 𝑂(ℎ1.00) and 𝑂(ℎ0.76) in the same settings. Globally optimal 15-day routes for an Adriatic Sea mission are computed in under one minute on a standard workstation, confirming operational feasibility. This study presents a direct comparison of Eulerian and Lagrangian numerical formulations for Zermelo’s navigation problem and analyses their convergence behaviour within the viscosity solution framework. The results provide practical guidance for numerical method selection and grid resolution in time-optimal ship routing problems.