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

Vahit Çalışır

doi:10.33714/masteb.1900720

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

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

Supporting Institution

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

Ethical Statement

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

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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Details

Primary Language

English

Subjects

Ocean Engineering

Journal Section

Research Article

Authors

Vahit Çalışır ^*
0000-0001-6575-8988
Türkiye

Publication Date

March 31, 2026

Submission Date

March 2, 2026

Acceptance Date

March 30, 2026

Published in Issue

Year 2026 Volume: 15 Number: 1

DOI

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

IZ

https://izlik.org/JA43XA79WZ

Cite

RIS / Bibtex

APA

Çalışır, V. (2026). Numerical solutions to Zermelo’s navigation problem under variable ocean current fields. Marine Science and Technology Bulletin, 15(1), 37-52. https://doi.org/10.33714/masteb.1900720

AMA

1.Çalışır V. Numerical solutions to Zermelo’s navigation problem under variable ocean current fields. Mar. Sci. Tech. Bull. 2026;15(1):37-52. doi:10.33714/masteb.1900720

Chicago

Çalışır, Vahit. 2026. “Numerical Solutions to Zermelo’s Navigation Problem under Variable Ocean Current Fields”. Marine Science and Technology Bulletin 15 (1): 37-52. https://doi.org/10.33714/masteb.1900720.

EndNote

Çalışır V (March 1, 2026) Numerical solutions to Zermelo’s navigation problem under variable ocean current fields. Marine Science and Technology Bulletin 15 1 37–52.

IEEE

[1]V. Çalışır, “Numerical solutions to Zermelo’s navigation problem under variable ocean current fields”, Mar. Sci. Tech. Bull., vol. 15, no. 1, pp. 37–52, Mar. 2026, doi: 10.33714/masteb.1900720.

ISNAD

Çalışır, Vahit. “Numerical Solutions to Zermelo’s Navigation Problem under Variable Ocean Current Fields”. Marine Science and Technology Bulletin 15/1 (March 1, 2026): 37-52. https://doi.org/10.33714/masteb.1900720.

JAMA

1.Çalışır V. Numerical solutions to Zermelo’s navigation problem under variable ocean current fields. Mar. Sci. Tech. Bull. 2026;15:37–52.

MLA

Çalışır, Vahit. “Numerical Solutions to Zermelo’s Navigation Problem under Variable Ocean Current Fields”. Marine Science and Technology Bulletin, vol. 15, no. 1, Mar. 2026, pp. 37-52, doi:10.33714/masteb.1900720.

Vancouver

1.Vahit Çalışır. Numerical solutions to Zermelo’s navigation problem under variable ocean current fields. Mar. Sci. Tech. Bull. 2026 Mar. 1;15(1):37-52. doi:10.33714/masteb.1900720