Trigonometrik B-spline kollokasyon sonlu eleman yöntemiyle düzenli uzun dalga (RLW) denkleminin nümerik çözümleri
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Dosyalar
Tarih
2020
Yazarlar
Dergi Başlığı
Dergi ISSN
Cilt Başlığı
Yayıncı
İnönü Üniversitesi
Erişim Hakkı
info:eu-repo/semantics/openAccess
Özet
Bu yüksek lisans tezi beş bölümden oluşmaktadır. Birinci bölümde, tezin temel amacından kısaca bahsedildi ve dalga kavramı hakkında bilgi verildi. Ayrıca bu bölümde tezde göz önüne alınacak olan Düzenli Uzun Dalga (RLW) denkleminin tarihsel gelişiminden bahsedildi. İkinci bölümde, bu tez çalışmasında kullandığımız temel kavramlar olan sonlu elemanlar yöntemi, kollokasyon yöntemi ve kübik trigonometrik B-spline fonksiyonlar hakkında bilgi verildi. Üçüncü bölümde, RLW denklemi ile ilgili literatürdeki çalışmalar sunuldu ve nümerik çözümü yapılacak olan RLW denkleminin farklı sınır ve başlangıç şartları ile verilen tek dalga hareketi, iki soliter dalga girişimi, ardışık dalga gelişimi ve dalga oluşumu problemleri tanıtıldı. Dördüncü bölümde, RLW denklemindeki lineer olmayan UUx terimi yerine beş farklı lineerleştirme tekniği kullanılarak kübik trigonometrik kollokasyon sonlu eleman yöntemi ile nümerik şemaları elde edildi ve önceki bölümde verilen model problemlere uygulandı. Elde edilen nümerik çözümler, çizelgeler halinde sunuldu ve mevcut tam çözüm ile literatürdeki farklı çalışmalardaki sonuçlarla karşılaştırıldı. Ayrıca tüm lineerleştirme teknikleri ile elde edilen nümerik şemaların kararlılık analizleri benzer olacağından sadece LİN-1 ile elde edilen şemanın kararlılık analizi von Neumann yöntemiyle incelendi. Son olarak beşinci bölümde, uygulanan tüm lineerleştirme tekniklerinden elde edilen sonuçlar her bir model problem için kendi içerisinde çizelgeler halinde karşılaştırılarak öne çıkan yöntemler sunuldu.
This master thesis consists of five chapters. In the first chapter, the main purpose of the thesis has been briefly mentioned and some information has been given about the wave concept. Also, in this chapter, the historical development of the Regular Long Wave (RLW) equation, which is going to be taken into consideration in the thesis, is mentioned. In the second chapter, some fundamental information about the basic concepts that we use in this thesis namely the finite element method, the collocation method and the cubic trigonometric B-spline functions are presented. In the third chapter, studies in the literature on the RLW equation are presented and the problems of single wave motion, two solitary wave interaction, sequential wave evoluation and undular bore devolopment problems given with different boundary and initial conditions of the RLW equation whose numerical solution will be made are introduced. In the fourth chapter, numerical schemes of cubic trigonometric collocation with finite element method are obtained by using five different linearization techniques instead of the nonlinear UUx term in the RLW equation, and the model given in the previous section was applied to the problems. The numerical solutions obtained were presented in tables and compared with the results of different studies in the literature with the exact solution. In addition, since the stability analysis of the numerical schemes obtained by all linearization techniques would be similar, the stability analysis of the scheme obtained only with L˙IN-1 was examined by von Neumann method. Finally, in the fifth chapter, the results obtained from all applied linearization techniques are compared for each model problem in tables within itself and prominent methods which are presented.
This master thesis consists of five chapters. In the first chapter, the main purpose of the thesis has been briefly mentioned and some information has been given about the wave concept. Also, in this chapter, the historical development of the Regular Long Wave (RLW) equation, which is going to be taken into consideration in the thesis, is mentioned. In the second chapter, some fundamental information about the basic concepts that we use in this thesis namely the finite element method, the collocation method and the cubic trigonometric B-spline functions are presented. In the third chapter, studies in the literature on the RLW equation are presented and the problems of single wave motion, two solitary wave interaction, sequential wave evoluation and undular bore devolopment problems given with different boundary and initial conditions of the RLW equation whose numerical solution will be made are introduced. In the fourth chapter, numerical schemes of cubic trigonometric collocation with finite element method are obtained by using five different linearization techniques instead of the nonlinear UUx term in the RLW equation, and the model given in the previous section was applied to the problems. The numerical solutions obtained were presented in tables and compared with the results of different studies in the literature with the exact solution. In addition, since the stability analysis of the numerical schemes obtained by all linearization techniques would be similar, the stability analysis of the scheme obtained only with L˙IN-1 was examined by von Neumann method. Finally, in the fifth chapter, the results obtained from all applied linearization techniques are compared for each model problem in tables within itself and prominent methods which are presented.
Açıklama
Anahtar Kelimeler
Matematik, Mathematics
Kaynak
WoS Q Değeri
Scopus Q Değeri
Cilt
Sayı
Künye
Yıldırım, H. (2020). Trigonometrik B-spline kollokasyon sonlu eleman yöntemiyle düzenli uzun dalga (RLW) denkleminin nümerik çözümleri. Yayınlanmış Yüksek Lisans Tezi, İnönü Üniversitesi.
