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

Yıl 2024, Cilt: 24 Sayı: 5, 1120 - 1128, 01.10.2024

İhsan Çelikkaya

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

Öz

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

Anahtar Kelimeler

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

Proje Numarası

yok

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

Öz

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.

Anahtar Kelimeler

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

Proje Numarası

yok

Kaynakça

• Bateman, H., 1915. Some recent researches on the motion of the fluids. Monthly Weather Review, 26, 163-170. Burgers, J. M., 1948. A mathematical model illustrating the theory of turbulence. Advances in applied mechanics, 1, 171-199. Brezis, H. and Browder, F., 1998. Partial differential equations in the 20th century. Advances in Mathematics, 135, 76-144.
• Cole, J. D., 1951. On a quai-linear parabolic equation occurring in aerodynamics. Quarterly of applied mathematics, 9, 225-236.
• Gao, y., Le, L.H. and Shi, B.C., 2013. Numerical solution of Burgers equation by lattice Boltzmann method. Applied mathematics and computation, 219, 7685-7692. http://dx.doi.org/10.1016/j.amc.2013.01.056
• Dag, İ., Irk, D. and Saka, B., 2005. A numerical solution of the Burgers equation using cubic B-splines. Applied mathematics and computation, 163, 199-211. https://doi.org/10.1016/j.amc.2004.01.028
• Saka, B. and Dağ, İ., 2007. Quartic B-spline collocation method to the numerical solution of the Burgers equation. Chaos, Solitons and Fractals, 32, 1125-1137. https://doi.org/10.1016/j.chaos.2005.11.037
• Kutluay, S. and Esen, A., 2004. A lumped Galerkin method for solving the Burgers equation. International journal of computer mathematics, 81, 1433-1444. https://doi.org/10.1080/00207160412331286833
• Dag, İ., Hepson, O.E. and Kacmaz, O., 2017. The trigonometric cubic B-spline algorithm for Burgers equation. International journal of nonlinear science, 24, 120-128.
• Ucar, y., Yagmurlu, N.M. and Celikkaya, İ., 2020. Numerical solution of Burgers type equation using finite element collocation method with Strang splitting. Mathematical Sciences and Applications E-Notes, 8, 29-45. https://doi.org/10.36753/mathenot.598635
• Dag, İ., Irk, D. and Sahin, A., 2004. B-spline collocation method for numerical solutions of the Burgers equation. Mathematical Problems in Engineering, 2005, 521-538. https://doi.org/10.1155/MPE.2005.521
• Mittal, R.C. and Jain, R.K., 2012. Numerical solutions of nonlinear Burgers’ equation with modified cubic B-splines collocation method. Applied Mathematics and Computation, 218, 7839-7855. https://doi.org/10.1016/j.amc.2012.01.059
• Ersoy, O., Dag, I. and Adar, N., 2018. Exponential twice continuously differentiable B-spline algorithm for Burgers equation. Ukrainian Mathematical Journal, 70, 788-800. https://doi.org/10.1007/s11253-018-1541-9
• Celikkaya, I. and Guzel, A., 2023. Four numerical schemes for solutions of Burgers equation via operator splitting trigonometric cubic B-spline collocation method. Journal of Applied Analysis and Computation, 13, 313-328. https://doi.org/10.11948/20220095
• Hundsdorfer, W., 2000. Numerical Solution of Advection-Diffusion-Reaction Equations. Lecture notes for PH.D. Course, Thomas Stieltjes Institute, Amsterdam.
• Creutz, M. and Gocksch, A., 1989. Higher-order hybrids Monte Carlo algorithms. Physics Letters A, 63, 9-12.
• Yoshida, H., 1990. Construction of higher order symplectic integrators. Physics Letters A, 150, 262-268. Sari, M., Tunc, H. and Seydaoglu, M., 2019. Higher order splitting approaches in analysis of the Burgers equation. Kuwait journal of science, 46, 1-14.
• Trotter, H.F., 1959. On the product of semi-groups of operators. Proceedings of the American Mathematical Society, 10, 545-551.
• Strang, G., 1968. On The Construction and Comparison Of Difference Schemes. SIAM Journal on Numerical Analysis, 5, 506-517.
• McCartin, B.J., 1991. Theory of exponential splines. Journal of approximation theory, 66, 1-23. Von Neumann, J. and Richtmyer, R.D., 1950. A Method for the Numerical Calculation of Hydrodynamic Shocks. Journal of Applied Physics, 21, 232-237.
• Asaithambi, A., 2010. Numerical solution of the Burgers’ equation by automatic differentiation. Applied Mathematics and Computation, 216, 2700-2708. https://doi.org/10.1016/j.amc.2010.03.115