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Development of Numerical Models for 1D and 2D Pollutant Transport Problems Using the Leapfrog Method with Additional Numerical Filters Hafiyyan, Qalbi; Nirmala, Azwa; Murad, Murad; Bachtiar, Vivi; Sumiyattinah, Sumiyattinah
Indonesian Journal of Multidisciplinary Science Vol. 5 No. 7 (2026): Indonesian Journal of Multidisciplinary Science
Publisher : International Journal Labs

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.55324/ijoms.v5i7.1277

Abstract

Pollutant transport in aquatic environments has become a critical environmental issue due to increasing contamination from industrial, domestic, and agricultural activities, which threaten ecosystems and human health. Understanding pollutant behavior through numerical modeling is essential for effective monitoring and mitigation strategies. This study aims to develop and evaluate a numerical model for simulating one- and two-dimensional pollutant transport using the advection–diffusion equation as the governing framework. The method employed is a finite difference approach based on the Leapfrog scheme combined with the Hansen numerical filter to improve model stability and simplicity. The model is tested through several benchmark cases, including pure advection, advection–diffusion, and Gaussian pulse scenarios in both one- and two-dimensional domains, with results compared to analytical solutions. The findings indicate that the proposed Leapfrog–Hansen model demonstrates good agreement with analytical solutions and achieves relatively small error values, particularly in one-dimensional cases. The results also show that spatial and temporal discretization significantly influence model stability and accuracy, where smaller step sizes generally improve performance but may reduce accuracy due to excessive filtering. In conclusion, the Leapfrog–Hansen model provides a simple yet effective alternative for pollutant transport simulation, especially for one-dimensional problems, with potential applications in environmental analysis and decision-making
A Stabilized Modified Lax Scheme with Numerical Filter for Accurate One-Dimensional Pollutant Transport Modeling Qalbi Hafiyyan; Azwa Nirmala
JURNAL TEKNIK SIPIL CENDEKIA (JTSC) Vol 7 No 2 (2026): May
Publisher : Departement of Civil Engineering, Universitas Winaya Mukti

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.51988/jtsc.v7i2.546

Abstract

Pollutant transport in water is a global issue commonly modeled using advection-diffusion equations that describe transport driven by concentration gradients and flow velocity. Because analytical solutions are limited to simple cases, numerical methods are essential for simulating pollutant transport. However, many existing numerical models require complex formulations to achieve accurate, stable solutions. This study proposes a one-dimensional numerical model based on a modified Lax scheme combined with the Hansen filter, providing stable, accurate solutions with a simpler formulation. The proposed model is evaluated using three test cases: pure advection, advection-diffusion, and a Gaussian pulse. The performance of the proposed numerical model is compared with the exact solution using L2, L?, and absolute error analysis. For the pure advection case, the proposed model achieves L2=0.048 and L?=0.011, which are lower than those reported by some previous numerical models. In the advection-diffusion case, the model also has better accuracy than some previous numerical models with L2=0.0948 and L?=0.0422. For the Gaussian pulse case, the absolute error remains very small at 8.76 × 10-5 at the concentration peak. show that the proposed model can suppress numerical oscillations while maintaining high accuracy and efficiency, making it effective for one-dimensional simulations of pollutant transport.
Development of Leapfrog-Hansen Numerical Model to Simulate One-Dimensional Dam-Break Flow Qalbi Hafiyyan; Irene Anggraini; Puspita Rahmasari; Azwa Nirmala
JTAM (Jurnal Teori dan Aplikasi Matematika) Vol 10, No 3 (2026): July
Publisher : Universitas Muhammadiyah Mataram

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.31764/jtam.v10i3.38595

Abstract

This study is a computational study aimed at developing and validating a numerical model for simulating one-dimensional dam-break flow. Dam construction, in addition to providing benefits to the community in terms of flood control, irrigation, and clean water sources, also carries the potential for disaster. Disasters can occur when dams fail or collapse. Dam failures can generate catastrophic flooding that threatens infrastructure, the environment, and human life. Therefore, numerical modeling is an important approach for understanding the characteristics of dam-break flow to support flood mitigation efforts following dam failure. The proposed model is developed using the Leapfrog finite-difference method, known for its simplicity. However, the conventional Leapfrog method is prone to numerical oscillations when handling discontinuities and shock waves in dam-break simulations. The novelty of this research lies in the development of a Leapfrog–Hansen numerical model for one-dimensional dam-break flow simulation by integrating the Hansen numerical filter into the conventional Leapfrog finite-difference scheme to improve stability while maintaining computational simplicity. The governing shallow water equations were solved using the proposed Leapfrog–Hansen model and applied to several hypothetical one-dimensional dam-break scenarios with varying downstream water depths. The performance of the developed model was evaluated by comparing its numerical simulation results with Stoker's analytical solution, which is often used as a benchmark in numerical modeling of one-dimensional dam-break flow. The comparison results show that the Leapfrog–Hansen model accurately reproduces the water surface profiles predicted by the analytical solution. The Leapfrog-Hansen model yielded relatively small Mean Absolute Error (MAE) values of 0.032 to 0.062, indicating high accuracy in reproducing dam-break flows. In addition, the developed model successfully reduces numerical oscillations in the conventional Leapfrog scheme and accurately captures flow discontinuities, shock-wave propagation, and wet-dry conditions, while maintaining simulation stability. These findings demonstrate that the proposed Leapfrog–Hansen model provides a simple, stable, and accurate alternative for simulating one-dimensional dam-break flows and has potential applications in flood-propagation analysis, preliminary dam-break hazard assessment, and other hydraulic studies related to flood risk mitigation.
APLIKASI METODE LEAST SQUARE PADA PEMODELAN PASANG SURUT DI PERAIRAN PEMANGKAT Qalbi Hafiyyan; Puspita Rahmasari; Wiwik Wiharti; Putri Yusella; Riza Aulia; Daslin S.J.L. Tobing
Jurnal Ilmiah Kurva Teknik Vol. 15 No. 1 (2026): Jurnal Ilmiah Kurva Teknik
Publisher : Program Studi Teknik Sipil Fakultas Teknik Universitas Mahasaraswati Denpasar

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.36733/jikt.v15i1.12530

Abstract

Perairan Pemangkat yang terletak di Kalimantan Barat merupakan kawasan pesisir dengan berbagai aktivitas seperti perikanan, pelayaran, dan pembangunan infrastruktur. Aktivitas-aktivitas tersebut sangat rentan dipengaruhi oleh pasang surut. Dengan demikian, pengetahuan dan pemahaman karakteristik pasang surut pada perairan pemangkat menjadi sangat penting. Pendekatan yang dapat dilakukan untuk memahami karakteristik pasang surut adalah pemodelan pasang surut dengan metode least square. Pendekatan ini umum digunakan karena mampu memberikan hasil yang akurat. Pada penelitian ini, pasang surut dimodelkan berdasarkan data observasi pasang surut sepanjang 15 hari dengan luaran berupa amplitudo dan fase 9 komponen harmonik. Hasil pemodelan menunjukkan bahwa komponen harmonik M2 menjadi komponen harmonik dengan nilai tertinggi. Hal tersebut menunjukkan bahwa efek gravitasi bulan sangat mempengaruhi karakteristik pasang surut perairan Pemangkat. Selain itu, penelitian ini juga memberikan nilai bilangan formzahl sebesar 0,955. Hal tersebut menunjukkan bahwa tipe pasang surut perairan Pemangkat adalah campuran cenderung harian ganda, dimana perairan mengalami pasang dan surut masing-masing dua kai dengan ketinggian yang berbeda. Selanjutnya, amplitudo dari komponen harmonik juga digunakan untuk menghitung elevasi muka air penting meliputi MHWL, HHWL, MSL, LHWL, dan LLWL. Adapun, nilai dari kelima elevasi tersebut adalah -0,525 m untuk LLWL, -0,329 m untuk MHWL, 0,555 m untuk MSL, 1,439 m untuk MHWL, dan 1,635 m untuk HHWL. Akhirnya, penelitian ini diproyeksikan bisa menjadi referensi dalam merencanakan berbagai aspek di perairan pemangkat, seperti pelayaran dan infrastruktur.