A Study on Grid-Size and Time-Step Calculation using the Taylor Series in Time Series Water Wave Modeling

Abstract

This paper presents the time-step and grid-size calculation methods for solving time series wave differential equations using the Finite Difference Method.The methods are formulated using the Taylor series, assuming that at a very small time-step and grid-size, the number of terms in the Taylor series starting with 2nd order with the highest order is much smaller than the order 1. I.

Alternative Algorithms for Time Series Water Wave Modeling

“…A weighted Taylor series is a truncated Taylor series limited to first-order terms, with the influence of higher-order terms encapsulated in coefficients known as weighting coefficients. The updated weighting coefficient values are presented in Table (1), wherein these coefficients are dependent on the optimization coefficient 𝜀 . A larger 𝜀 corresponds to a greater influence of higher-order Taylor series terms.…”

“…The methodology for computing these weighting coefficients is detailed in Hutahaean (2023). However, the precision of the weighting coefficients listed in Table (1) surpasses that of Hutahaean (2023). In this particular time series model, the selection of weighting coefficients hinges on the wave amplitude.…”

“…Fig (1). Depth average velocity concept Equation (4) was initially formulated to model very long waves, such as tidal waves, using a horizontal velocity integration coefficient 𝛽 𝑢 = 1.…”

“…

This research builds upon Hutahaean (2024a), focusing on time series water wave modeling, with substantial portions of this article derived from that prior study. Modifications were made to the calculation algorithm; specifically, vertical water particle velocity is now computed using the continuity equation, and water surface elevation is determined through kinematic free surface boundary conditions.

…”