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Journals / Sinop Üniversitesi Fen Bilimleri Dergisi / 2026 / Cilt: 11 Sayı: 1

Numerical Solution of GRLW Equation by Cubic Hermit B-Spline Collocation Method

GRLW Denkleminin Kübik Hermite B-splayn Kollakasyon Yöntemiyle Sayısal Çözümü

Sibel Özer(İNÖNÜ ÜNİVERSİTESİ)0000-0003-4956-4002
Yusuf Uçar(İNÖNÜ ÜNİVERSİTESİ)0000-0003-1469-5002
Halil Beyazıt(İnönü Üniversitesi Fen Bilimleri Enstitüsü)0000-0002-0854-2301
Pages
401–423
Publication Date
2026-06-26
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AI Summary & Key Takeaways

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

The tables and graphs show that the proposed numerical scheme sufficiently preserves the conservation constants of the problems and is in good agreement with the existing studies.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:This study deals with the numerical solution of the Generalized Regular Long Wave (GRLW) equation, which holds an important place in fluid dynamics.
📊
Key Findings & Data:To this end, first of all, the GRLW equation is discretized by using the Crank-Nicolson method, then linearized using the Rubin-Graves linearization technique for nonlinear terms, and a linear numerical scheme is obtained by applying the cubic Hermite B-spline collocation finite element method.
💡
Conclusion & Impact:The tables and graphs show that the proposed numerical scheme sufficiently preserves the conservation constants of the problems and is in good agreement with the existing studies.
🔬Design:Araştırma Makalesi
📍Scope:Comprehensive Sample
⚙️Variables:GRLW denklemi · kollakasyon yöntemi · kübik Hermite B-splayn

Abstract

This study deals with the numerical solution of the Generalized Regular Long Wave (GRLW) equation, which holds an important place in fluid dynamics. To this end, first of all, the GRLW equation is discretized by using the Crank-Nicolson method, then linearized using the Rubin-Graves linearization technique for nonlinear terms, and a linear numerical scheme is obtained by applying the cubic Hermite B-spline collocation finite element method. This proposed numerical scheme is applied to a solitary wave problem and the interaction of two and three solitary waves. Conservation constants and the absolute change quantities of the conservation constants are calculated for all problems, the error norms for a single solitary wave problem which has an exact analytical solution. The calculated error norms and the conservation constants are presented in tables and compared with those of the existing studies for the same parameters to test the accuracy of the method. Furthermore, the graphs of the obtained numerical results are drawn to visually present the physical behavior of the problems. The tables and graphs show that the proposed numerical scheme sufficiently preserves the conservation constants of the problems and is in good agreement with the existing studies.

Özet

Bu çalışma, akışkanlar dinamiğinde önemli bir yere sahip olan Genelleştirilmiş Düzenli Uzun Dalga (GRLW) denkleminin sayısal çözümünü içermektedir. Bu amaçla, GRLW denklemi Crank-Nicolson yöntemi ile ayrıklaştırıldıktan sonra, doğrusal olmayan terim Rubin-Graves doğrusallaştırma tekniğiyle lineerleştirildi ve kübik Hermite B-splayn kollakasyon sonlu elemanlar yöntemi uygulanarak bir lineer sayısal şema elde edildi. Bu önerilen sayısal şema bir soliter dalga problemi ile iki ve üç soliter dalganın girişimi problemlerine uygulanarak korunum sabitleri ve korunum sabitlerinin mutlak değişim miktarları ve ayrıca analitik çözümü mevcut olan bir soliter dalga problemi için hata normları hesaplandı. Hesaplanan hata normları ve korumum sabitleri tablolar halinde sunuldu ve aynı parametreler için mevcut olan çalışmalarla karşılaştırılarak yöntemin doğruluğu test edildi. Ayrıca, elde edilen sayısal sonuçların grafikleri çizilerek problemlerin fiziksel davranışları görsel olarak sunuldu. Tablolar ve grafikler, önerilen sayısal şemanın problemlerin korunum sabitlerini yeterince iyi bir şekilde koruduğunu ve mevcut olan çalışmalarla oldukça uyumlu olduğunu göstermektedir.

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Cite This Article

Özer, S., Uçar, Y., Beyazıt, H. (2026). Numerical Solution of GRLW Equation by Cubic Hermit B-Spline Collocation Method. Sinop Üniversitesi Fen Bilimleri Dergisi, Cilt: 11 Sayı: 1, 401–423. https://doi.org/10.33484/sinopfbd.1832547

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