Pemodelan Numerik dan Validasi Kuantitatif Persamaan Diferensial Nonlinier untuk Sistem Aliran Air Berkelanjutan

Authors

Binti Karomah Universitas Surakarta
Ferdinand Hasudungan Siagian
A. Eka Hermia Fitrianingsy

DOI:

Keywords:

Persamaan diferensial nonlinier, Runge-Kutta orde empat, pemodelan aliran air, validasi kuantitatif, keberlanjutan, DAS Bengawan Solo

Abstract

This study aims to construct a nonlinear differential equation model based on mass balance to represent the relationships among inflow, storage, outflow, and water loss and to evaluate the accuracy of its numerical solution. The method employed a quantitative simulation using controlled synthetic data, with the Bengawan Solo Watershed serving as the contextual setting for model implementation. The dataset consisted of 30 observations over 174 hours at six-hour intervals. The differential equation was solved using the fourth-order Runge-Kutta (RK4) method with a 0.25-hour time step. Validation was conducted for water level and outflow discharge using RMSE, MAE, MAPE, coefficient of determination (R²), Nash-Sutcliffe Efficiency (NSE), Kling-Gupta Efficiency (KGE), Shapiro-Wilk test, and paired t-test. The results show that the simulation reproduced the pseudo-observation patterns with low errors. The RMSE was 0.0091 m for water level and 0.5815 m³/h for discharge. The R², NSE, and KGE values reached 0.9976, 0.9976, and 0.9962 for water level and 0.9952, 0.9949, and 0.9866 for discharge, respectively. Discharge MAPE was 1.64%, while the paired t-test indicated no significant mean difference between the simulation and pseudo-observations. The system maintained a minimum discharge of 20 m³/h for 90% of the observation periods. The study concludes that the combination of a nonlinear mass-balance model, RK4, multi-metric validation, and reliability assessment provides a transparent quantitative framework for preliminary testing of water-flow systems, while field validation remains necessary because the present results are based on synthetic data.

Keywords: Nonlinear Differential Equations; Fourth-Order Runge-Kutta; Water Flow Modeling; Quantitative Validation; Sustainability; Bengawan Solo Watershed

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