Determination of Time Step and Grid Size Based on the Formulation Conditions of the Fundamental Hydrodynamic Equations for Time-Series Numerical Water Wave Modeling

Syawaluddin Hutahaean

conditions for formulating the basic equations of hydrodynamics.

The formulation of the fundamental hydrodynamic equations is based on two key assumptions. The first arises from the derivation of the Euler momentum conservation equation, which assumes that the temporal and spatial increments (dt, dx) are sufficiently small for the Taylor series expansion to be truncated after the first-order terms. The second assumption is associated with the continuity equation, in which the fluid-particle velocity is assumed to remain constant over the small time interval dt. Based on these two assumptions, two optimization equations are formulated with two unknowns, time step and grid size (dt, dx). Solving these equations provides the optimal time step and grid size for numerical simulations. The resulting values are consistent with the assumptions underlying the derivation of the fundamental hydrodynamic equations while also satisfying the requirement of sufficiently small discretization intervals imposed by numerical methods. The proposed approach enables more accurate numerical simulations of challenging wave conditions, particularly during wave breaking and post-breaking propagation.

Received: 15 June 2026, Received in revised form: 17 July 2026, Accepted: 21 July 2026, Available online: 26 July 2026

10.22161/ijaers.137.5