Rosenau-RLW denkleminin çözümü için sonlu fark yöntemi üzerine temellenmiş bir nümerik şema
Küçük Resim Yok
Tarih
2023
Yazarlar
Dergi Başlığı
Dergi ISSN
Cilt Başlığı
Yayıncı
İnönü Üniversitesi
Erişim Hakkı
info:eu-repo/semantics/openAccess
Özet
Bu tezin içeriği şu şekilde düzenlenmiştir: Bölüm 1 de tezde ele alınacak olan 1-boyutlu lineer olmayan genel Rosenau-RLW denklemi üzerine özellikle nümerik anlamda yapılan çalışmalarla ilgili bir literatür taraması verildi. Bir sonraki bölümde yani Bölüm 2 de önce bu çalışmada sunulacak olan nümerik ¸semanın temelini oluşturan türevlere fark yaklaşımlarından ve sonlu fark yönteminin bir başlangıç ve sınır değer problemine uygulanışındaki temel prensiplerle birlikte tutarlılık, kararlılık, yakınsaklık ve Lax'ın denklik teoreminden kısaca bahsedildi. Sonra, model problem olarak göz önüne alınan 1-boyutlu lineer olmayan genel Rosenau-RLW denklemiyle verilen başlangıç ve sınır değer probleminin sonlu farklar yöntemine dayanan bir nümerik ¸semasının çıkarılmasına geçildi. Bunun için önce denklemde görülen zaman yönündeki türevler yerine birinci mertebeden hata terimli standart ileri fark formülü ve konum yönündeki türevler yerine de Crank-Nicolson tipi yaklaşımı yazılarak sadece zaman yönünde ayrıklaştırılmı¸s fark denklemi bulundu. Bundan sonra, konum yönünde ayrıklaştırma için, denklemdeki lineer olmayan terim yerine zamana göre ikinci mertebeden hata terimli Rubin-Graves tipi lineerleştirme yaklaşımı ve konuma göre tüm türevler yerine de ikinci mertebeden hata terimli standart merkezi fark formülleri yazılarak lineer cebirsel denklem sistemiyle sonuçlanan ve kolayca uygulanabilen tamamen ayrıklaştırılmış Crank-Nicolson tipi fark ¸seması elde edildi. Bölüm 3 de ise sunulan ¸sema, model problemde görülen parametrelerin bazı özel durumları için seçilen örnek problemlere uygulandı. Şemanın doğruluğu ve güvenirliği ile birlikte mevcut yöntemin yakınsama oranı analizini desteklediğini göstermek için elde edilen yaklaşık nümerik sonuçlar (çözümün noktasal değerleri, ortalama ve maksimum hata normları, kütle ve enerji korunum sabitleri, ¸semanın yakınsaklık mertebesi) analitik ve diğer araştırmacıların buldukları sonuçlarla karşılaştırmalı olarak çizelgeler halinde verildi. Ayrıca, mevcut çözümlerin sürekliliğini ve problemin doğru fiziksel davranışlarını sergilediğini doğrulamak için bazı grafikler çizildi. Son olarak Bölüm 4 de kısa bir sonuçla birlikte ileriye yönelik çalışmalardan bahsedilerek bu tez çalışması sonlandırıldı. Anahtar Kelimeler: Rosenau-RLW Denklemi, Sonlu Farklar, Crank- Nicolson Tipi Yaklaşım, Rubin-Graves Tipi Lineerleştirme Tekniği, Kararlılık Testi
The content of this thesis is organized as follows: In Chapter 1, a literature review is given, especially regarding the numerical studies on the 1-dimensional nonlinear general Rosenau-RLW equation, which will be discussed in the thesis. In the next chapter, that is, Chapter 2, the difference approaches to derivatives that form the basis of the numerical scheme to be presented in this study, and the basic principles in the application of the finite difference method to an initial and boundary value problem, as well as consistency, stability, convergence and Lax's equivalence theorem, are briefly mentioned. Then, a numerical diagram based on the finite difference method was created for the initial and boundary value problem given by the 1-dimensional nonlinear general Rosenau-RLW equation, which is considered as a model problem. For this purpose, first, the standard forward difference formula with first-order error term was written instead of the derivatives in the time direction seen in the equation, and the Crank-Nicolson type approach was written instead of the derivatives in the position direction, and a difference equation discretized only in the time direction was found. After this, for the discretization in the position direction, the Rubin-Graves type linearization approach with a second-order error term with respect to time was written instead of the nonlinear term in the equation, and standard central difference formulas with a second-order error term were written instead of all derivatives with respect to position, resulting in a linear algebraic equation system that can be easily applied. A discretized Crank-Nicolson type difference diagram was obtained. In Chapter 3, the diagram presented was applied to sample problems selected for some special cases of the parameters seen in the model problem. The approximate numerical results obtained (point values of the solution, average and maximum error norms, mass and energy conservation constants, order of convergence of the scheme) are presented in analytical and comparative tables with the results found by other researchers, in order to show that the current method supports the convergence rate analysis, as well as the accuracy and reliability of the scheme. given as. Additionally, some graphs were drawn to verify the continuity of the existing solutions and that they exhibited the correct physical behavior of the problem. Finally, in Chapter 4, this thesis study was concluded by giving a short conclusion and mentioning future studies. Key Words: Rosenau-RLW Equation, Finite Differences, Crank-Nicolson Type Approach, Rubin-Graves Type Linearization Technique, Stability Test
The content of this thesis is organized as follows: In Chapter 1, a literature review is given, especially regarding the numerical studies on the 1-dimensional nonlinear general Rosenau-RLW equation, which will be discussed in the thesis. In the next chapter, that is, Chapter 2, the difference approaches to derivatives that form the basis of the numerical scheme to be presented in this study, and the basic principles in the application of the finite difference method to an initial and boundary value problem, as well as consistency, stability, convergence and Lax's equivalence theorem, are briefly mentioned. Then, a numerical diagram based on the finite difference method was created for the initial and boundary value problem given by the 1-dimensional nonlinear general Rosenau-RLW equation, which is considered as a model problem. For this purpose, first, the standard forward difference formula with first-order error term was written instead of the derivatives in the time direction seen in the equation, and the Crank-Nicolson type approach was written instead of the derivatives in the position direction, and a difference equation discretized only in the time direction was found. After this, for the discretization in the position direction, the Rubin-Graves type linearization approach with a second-order error term with respect to time was written instead of the nonlinear term in the equation, and standard central difference formulas with a second-order error term were written instead of all derivatives with respect to position, resulting in a linear algebraic equation system that can be easily applied. A discretized Crank-Nicolson type difference diagram was obtained. In Chapter 3, the diagram presented was applied to sample problems selected for some special cases of the parameters seen in the model problem. The approximate numerical results obtained (point values of the solution, average and maximum error norms, mass and energy conservation constants, order of convergence of the scheme) are presented in analytical and comparative tables with the results found by other researchers, in order to show that the current method supports the convergence rate analysis, as well as the accuracy and reliability of the scheme. given as. Additionally, some graphs were drawn to verify the continuity of the existing solutions and that they exhibited the correct physical behavior of the problem. Finally, in Chapter 4, this thesis study was concluded by giving a short conclusion and mentioning future studies. Key Words: Rosenau-RLW Equation, Finite Differences, Crank-Nicolson Type Approach, Rubin-Graves Type Linearization Technique, Stability Test
Açıklama
Anahtar Kelimeler
Matematik, Mathematics
