Sabit Katsayılı İki Boyutlu Lineer Hiperbolik Denklemler İçin Cauchy Probleminin Süreksiz Fonksiyonlar Sınıfında Yüksek Mertebeden Hassas Sonlu Farklar Şeması

Yıl 2020, Cilt: 13 Sayı: 1, 6 - 12, 30.06.2020

Öykü Yener , Bahaddin Sinsoysal , Mahir Resulov

https://doi.org/10.20854/bujse.736345

Öz

Bu çalışmada, hidrodinamiğin çeşitli alanlarında karşılaşılan iki boyutlu skaler adveksiyon denklemi için yazılmış Cauchy probleminin pratik hesaplanması için bir sonlu fark şeması geliştirilmiştir. Bu amaçla, ana probleme göre bazı avantajları olan bir yardımcı problem sunulmuştur. Önerilen yardımcı problem, daha yüksek mertebeden hassas bir sonlu farklar şeması oluşturmaya imkan sağlar.

Anahtar Kelimeler

Hidrodinamiğin model denklemleri, Süreksiz fonksiyonlar sınıfında zayıf çözüm

A Higher-Order Sensitive Finite Differences Scheme Of The Cauchy Problem For 2D Linear Hyperbolic Equations With Constant Coefficients In A Class Of Discontinuous Functions

Öz

In this study we develop a finite difference schema for practical calculation of the Cauchy problem for the 2D scalar advection equation with a higher accuracy oder constant coefficient, encountered in different areas of hydrodynamics. For this aim to develop an auxiliary problem having some advantages over the main problem is introduced. The proposed auxiliary problem permits us construct a higher order sensitive finite differences schema.

Anahtar Kelimeler

Modelling equations of hydrodynamics, Weak solution in a class of discontinuous functions

Kaynakça

• [1] Ames, W.F. Nonlinear Partial Differential Equations in Engineering, Academic Press, New York, London, 1965.
• [2] Ames, W.F. Numerical Methods for Partial Differential Equations. Academic Press, New York, 1977.
• [3] Anderson, D.A., Tannehill, J.C., Pletcher, R.H. Computational Fluid Mechanics and Heat Transfer, Vol. 1,2, Hemisphere Publishing Corporation, 1984.
• [4] Friedrichs, K.O. Nonlinear Hyperbolic Differential Equations for Functions of Two Independent Variables, American Journal of Mathematics, Vol. 70, pp.555-589, 1948.
• [5] Fritz, John. Partial Differential Equations, Springer-Verlag, New York, Heidelberg, Berlin, 1986.
• [6] Godunov, S.K., Ryabenkii, V.S. Finite Difference Schemes. Moskow, Nauka, 1972. [7] Godunov, S.K. Equations of Mathematical Physicis. Nauka, Moskow, 1979.
• [8] Goritskii, A.A., Krujkov, S.N., Chechkin, G.A. A First Order Ouasi-Linear Equations with Partial Differential Derivaites. Pub. Moskow University, Moskow, 1997.
• [9] LeVeque R.J. Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, 2002, 558p.
• [10] Noh, W.F., Protter, M.N. Difference Methods and the Equations of Hydrodynamics, Journal of Math. and Mechanics, Vol. 12, No. 2, 1963.
• [11] Rasulov, M.A. On a Method of Solving the Cauchy Problem for a First Order Nonlinear Equation of Hyperbolic Type with a Smooth Initial Condition, Soviet Math. Dok. 43, No.1, 1991.
• [12] Rasulov, M.A., Ragimova, T.A. A Numerical Method of the Solition of ne Nonlinear Equation of a Hyperbolic Type of the First Order Differential Equations, Minsk, Vol. 28, No.7, pp. 2056-2063, 1992.
• [13] Rasulov, M.A. Identification of the Saturation Jump in the Process of Oil Displacement by Water in a 2D Domain, Dokl RAN, Vol. 319, No.4, pp. 943-947, 1991.
• [14] Rasulov, M.A., Coskun, E., B. Sinsoysal. Finite Differences Method for a Two-Dimensional Nonlinear Hyperbolic Equations in a Class of Discontinuous Functions. App. Mathematics and Computation, vol.140, Issue l, August, pp.279-295, 2003, USA.
• [15] Richmyer, R.D., Morton, K.W. Difference Methods for Initial Value Problems, New York, Wiley, Int., 1967.
• [16] Samarskii, A.A. Theory of Difference Schemes. Moskow, Nauka, 1977.
• [17] Smoller, J.A. Shock Wave and Reaction Diffusion Equations, Springer-Verlag, New York Inc., 1983.
• [18] Toro, Eleuterio, F . Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer-Verlag, Berlin Heidelberg, 1999.
• [19] Whitham, G.B. Linear and Nonlinear Waves, Wiley Int., New York, 1974.