Linear advection equations are of great practical importance in the evolution of a variety of physical phenomena, in particular, hydrodynamic, thermal, and mass transfer processes that occur in the presence of ocean currents. The present work proposes an extension of the Path Tubes method, an explicit, backtracking-in-time, semi-Lagrangian, and mass-conserving algorithm, for solving a challenging advection problem based on Stommel’s ocean current model. The Path Tubes method equipped with an adaptative time step EDO-solving method increased its robustness and exhibited an accurate, non-oscillatory, and non-diffusive behavior.