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

Year 2026, Volume: 15 Issue: 1 , 37 - 52 , 31.03.2026

Vahit Çalışır

https://doi.org/10.33714/masteb.1900720

https://izlik.org/JA43XA79WZ

Abstract

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.

Keywords

Zermelo navigation problem , Hamilton-Jacobi-Bellman equation , WENO finite-difference scheme , Lagrangian extremal field , Optimal ship routing , Variable ocean currents

Ethical Statement

For this type of study, formal consent is not required.

Supporting Institution

This study was carried out without financial support from any institution.

Thanks

This work received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. Oceanographic velocity data for the Adriatic Sea benchmark were obtained from the Mediterranean Sea Physics Reanalysis product distributed by the Copernicus Marine Environment Monitoring Service (CMEMS), dataset identifier MEDSEA_MULTIYEAR_PHY_006_004; the specific mission scenario follows the configuration of Lolla et al. (2014). Computational experiments were performed on a standard workstation (Intel Core i7, 16 GB RAM) without dedicated HPC resources.

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