Burger Denklemi için Lie-Trotter ve Strang Splitting Operatör parçalama Yöntemine Dayalı Kübik Üstel B-spline kollokasyon Uygulaması

Year 2024, , 1120 - 1128, 01.10.2024

İhsan Çelikkaya

https://doi.org/10.35414/akufemubid.1464370

Abstract

Bu çalışmada, Burgers denkleminin nümerik çözümleri için kübik üstel B-spline kollokasyon ile birlikte operatör parçalama yöntemi önerildi. Operatör parçalama yöntemini uygulamak için Burgers denklemi zaman terimine göre lineer kısım (difüzyon) ve lineer olamayan kısım (konveksiyon) olarak iki alt denkleme parçalandı. Daha sonra her bir alt denkleme zaman yönünde Crank-Nicolson sonlu fark yaklaşımları, konum yönünde ise kübik üstel B-spline fonksiyonlarının ve türevlerinin x_m düğüm noktalarındaki değerleri uygulandı. Elde edilen cebirsel denklem sistemleri Lie-Trotter ve Strang parçalama şemaları kullanılarak ana denklemin nümerik çözümleri bulundu. Parçalama yöntemlerinin bazı avantajları çözümün fiziksel özelliklerini koruması, uzun zaman aralıklarında daha yakınsak sonuçlar vermesi, daha basit algoritmalara olanak sağlaması, çözüm vektörlerinin bilgisayarda depolanması olarak sayılabilir. Hesaplanan sayısal sonuçların doğruluğunu ölçmek için literatürde sıkça kullanılan L_2,L_∞ hata normları kullanıldı. Ayrıca elde edilen sonuçlar literatürdeki bazı çalışmalarla karşılaştırıldı. Uygulanan yöntemin kararlılık analizi von Neumann Fourier seri yöntemiyle incelendi

Keywords

Burgers Denklemi, Kübik Üstel B-spline, Kollokasyon Yöntemi, ; Lie-Trotter Parçalama, Strang Parçalama

Project Number

yok

Application of the Cubic Exponential B-spline Collocation Operator Splitting Method for Numerical solutions of Burgers Equation

Abstract

In this study, the cubic exponential B-spline collocation method has been proposed for the numerical solutions of the Burgers equation with the operator splitting. To apply the operator splitting method, the Burgers' equation has decomposed into two sub-equations based on the time term: the linear part (diffusion) and the nonlinear part (convection). Subsequently, for each sub-equation, Crank-Nicolson finite difference schemes in the temporal direction and cubic exponential B-spline functions and their derivatives, have applied at the x_m nodal points in the spatial direction. The algebraic equation systems obtained have been solved numerically using the Lie-Trotter and Strang splitting schemes to get the solutions of the main equation. Some advantages of the splitting methods include preserving the physical characteristics of the solution, yielding more convergent results over long time intervals, enabling simpler algorithms, and facilitating the storage of solution vectors on computer. To assess the accuracy of the computed numerical results the L_2 and L_∞ error norms have been used. Additionally, the obtained results have been compared with some studies in the literature. The stability analysis of the applied method has been investigated using the von Neumann Fourier series method.

Keywords

Burgers Equation, Exponential B-spline, Collocation Method, Lie-Trotter Splitting, Strang Splitting

Project Number

yok

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