An effective numerical technique for the Rosenau-KdV-RLW equation

Abstract

In this study, the Rosenau-Korteweg-de Vries-Regular Longwave (Rosenau-KdV-RLW) equation has been converted into a partial differential equation system consisting of two equations using a splitting technique.  Then, numerical solutions for the Rosenau-KdV-RLW equation system have been obtained using separately both cubic and quintic B-spline finite element collocation method.  For the unknowns in those equations, B-spline functions at x-position and Crank-Nicolson type finite difference approaches at time positions are used.  A test problem has been chosen to check the accuracy of the proposed discretized scheme.  The basic conservation properties of the Rosenau-KdV-RLW equation have been shown to be protected by the proposed numerical scheme.  The results are compared with the analytical solution of the problem and the results given in the literature.  For the reliability of the method the error norms L_2 and L_∞ are calculated.  It is seen that the proposed method gives harmonious results with exact solutions.

Keywords

• Rosenau-KdV-RLW,B-spline functions,collocation method,splitting technique

Rosenau-KdV-RLW denklemi için etkin bir sayısal teknik

Abstract

Bu çalışmada, Rosenau Korteweg-de Vries düzenli uzun dalga (Rosenau-KdV-RLW) denklemi bir parçalama tekniği kullanılarak iki denklemden oluşan bir kısmi diferansiyel denklem sistemine dönüştürülmüştür.  Daha sonra, Rosenau-KdV-RLW denklem sistemi için kübik ve kuintik B-spline sonlu eleman kollakasyon yöntemi kullanılarak sayısal çözümler önerilmiştir.  Bu denklemlerdeki bilinmeyenler için x-konumunda B-spline fonksiyonlar ve zaman konumunda Crank-Nicolson tipi sonlu fark yaklaşımları kullanılmıştır.  Önerilen sayısal şemaların doğruluğunu kontrol etmek için bir test problemi seçilmiştir.  Rosenau-KdV-RLW denkleminin temel korunum özelliklerinin önerilen sayısal şemalar ile korunduğu görülmüştür.  Elde edilen sonuçlar problemin analitik çözümü ve literatürde verilen sonuçlarla karşılaştırılmıştır.  Yöntemin güvenilirliği için L_2 ve L_∞ hata normları hesaplanmıştır.  Önerilen yöntemin tam çözümlerle uyumlu sonuçlar verdiği görülmüştür. 

Keywords

• Rosenau-KdV-RLW,B-spline fonksiyonlar,kollakasyon metodu,parçalama tekniği

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