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Journals / Black Sea Journal of Engineering and Science / 2026 / Cilt: 9 Sayı: 4

A Robust Method for the Numerical Solution of the Advection-Diffusion Equation Using the Galerkin Method Based on Hermite B-Spline Functions

Halil Beyazıt(İNÖNÜ ÜNİVERSİTESİ, FEN BİLİMLERİ ENSTİTÜSÜ, MATEMATİK (DR))0000-0002-0854-2301
Sibel Özer(İnönü Üniversitesi Fen Bilimleri Enstitüsü)0000-0003-4956-4002
Yusuf Uçar(İnönü Üniversitesi Fen Bilimleri Enstitüsü)0000-0003-1469-5002
Pages
1556–1567
Publication Date
2026-07-15
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Additionally, to observe the agreement of the approximate solutions with the analytical solution of the model problems, the graphs of the analytical and the approximate solutions were plotted together, and the graphs of the absolute error were plotted.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:The goal of this research is to find an approximate solution to the one-dimensional linear advection-diffusion equation using the cubic Hermite B-Spline Galerkin finite element method, a new numerical method developed by selecting cubic Hermite B-spline functions as the basis functions.
📊
Key Findings & Data:After discretizing the advection-diffusion equation in the temporal direction using the Crank-Nicolson formula, the spatial discretization of the semi-discrete scheme was performed using cubic Hermite B-spline functions.
💡
Conclusion & Impact:Additionally, to observe the agreement of the approximate solutions with the analytical solution of the model problems, the graphs of the analytical and the approximate solutions were plotted together, and the graphs of the absolute error were plotted.
🔬Design:Modeling & Simulation
📍Scope:Comprehensive Sample
⚙️Variables:Galerkin method · Advection-diffusion equation · Hermite B-spline function

Abstract

The goal of this research is to find an approximate solution to the one-dimensional linear advection-diffusion equation using the cubic Hermite B-Spline Galerkin finite element method, a new numerical method developed by selecting cubic Hermite B-spline functions as the basis functions. After discretizing the advection-diffusion equation in the temporal direction using the Crank-Nicolson formula, the spatial discretization of the semi-discrete scheme was performed using cubic Hermite B-spline functions. The effectiveness and accuracy of the proposed method were tested using the four most well-known model problems in the literature. The obtained numerical results were compared with those of other studies in the literature and presented in tabular form. Additionally, to observe the agreement of the approximate solutions with the analytical solution of the model problems, the graphs of the analytical and the approximate solutions were plotted together, and the graphs of the absolute error were plotted.

Özet

The goal of this research is to find an approximate solution to the one-dimensional linear advection-diffusion equation using the cubic Hermite B-Spline Galerkin finite element method, a new numerical method developed by selecting cubic Hermite B-spline functions as the basis functions. After discretizing the advection-diffusion equation in the temporal direction using the Crank-Nicolson formula, the spatial discretization of the semi-discrete scheme was performed using cubic Hermite B-spline functions. The effectiveness and accuracy of the proposed method were tested using the four most well-known model problems in the literature. The obtained numerical results were compared with those of other studies in the literature and presented in tabular form. Additionally, to observe the agreement of the approximate solutions with the analytical solution of the model problems, the graphs of the analytical and the approximate solutions were plotted together, and the graphs of the absolute error were plotted.

🔖

Cite This Article

Beyazıt, H., Özer, S., Uçar, Y. (2026). A Robust Method for the Numerical Solution of the Advection-Diffusion Equation Using the Galerkin Method Based on Hermite B-Spline Functions. Black Sea Journal of Engineering and Science, Cilt: 9 Sayı: 4, 1556–1567. https://doi.org/10.34248/bsengineering.1919861

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