İki-Boyutlu Konvektif Sınır Koşullu Erime Problemi İçin Nümerik Yaklaşım
Bu çalışmada, daha önce çözdüğümüz, iki-boyutlu konvektif sınır koşullu erime probleminde, türevlerin bir kısmında açık yöntem kullanırken bir kısmında da kapalı yöntem kullanarak sonlu farklar oluşturulmuştur ve bu denklemlerin çözümü için bir iteratif yöntem geliştirilmiştir. Metod (x, y) koordinatlarında ikinci dereceden doğruluğa sahiptir. Bu metodla elde edilen sonuçlar, önceki araştırmacılar tarafından verilen sonuçlarla tamamen uyumludur.
Numerical Approach for the two-dimensional heat equation problem with convective boundary conditions
In this work, we extended our earlier study on the solution of two-dimensional heat equation problem by considering a class of time-split finite difference methods. Operator splitting is used as a procedure for computing, some derivatives are computed explicitly and some of them computed implicitly during this procedure. The procedure is second order accurate in time and in (x, y) coordinates. The results of computing by present procedure are in totally compatible with the results obtained previously by other researches.
___
- T. Öziş , V. Gülkaç, “Application of variable
interchange method for solution of twodimensional
fusion problem with convective
boundary conditions,” Numerical Heat Transfer,
Part A, vol. 44, no. 1, 44, pp. 85-95, 2003.
- V. Gülkaç, “A Numerical Solution of TwoDimensional
Fusion Problem with Convective
Boundary Conditions,” International Journal for
Computational Methods in Engineering Science
and Mechanics, vol. 11, no.1, pp. 20-26, 2010.
- J. Crank, Numerical Methods in Heat Transfer,
John Wiley, 1981.
- T. R. Goodmann, Application of integral Methods
to Transient Non-Linear Heat Heat-Transfer, in T.
F. Irvine., Jr., and J. P. Harnett (eds.), Advances
in Heat Transfer, pp. 51-122. Academic Press,
New-York, 1964.
- H. Rasmussen, “An Approximate Method for
Solving Two-Dimensional Stefan Problems,”
Lett. Heat Mass Transfer, vol. 4, pp. 273-277,
1997.
- C. W. Cryer, A Survey of Steady-State Porous
Flow Free Boundary Problems, MRC Tech.
Summary Report 1657, University of Wisconsin,
Madison, 1976. vol.R. M. Furzeland , A Survey of
the Formulation and Solution of Free and Moving
Boundary Problems, Technical Report TR76,
Department of Mathematics, Brunel University,
London, England, 1977.
- R. M. Furzeland , “Symposium on Free and
Moving Boundary Problems in Heat Flow and
Diffusion,” Bull. Inst. Maths Applics, vol. 15, pp.
172-176, 1979.
- J. M. Aitchison, “Numerical Treatment of a
Singularity in a Free Boundary Problem,” Proceedings of the Royal Society of London,
Series A, pp. 573-580, 1972.
- J. M. Aitchison, “The Numerical Solution of a
Minimization Problem Associated with a Free
Surface Flow,” J. Inst. Maths Applics, vol. 20, pp.
33-44, 1977.
- H. G. Landau, “Heat Conduction in a Melting
Solid,” Qart. Appl. Math., vol. 8, pp. 81-94, 1950.
- R. S. Gupta, A. Kumar, “Isotherm Migration
Method Applied to Fusion Problems with
Convective Boundary Conditions,” Int. J. Numer.
Meth. Eng. vol. 26, pp. 2547-2558, 1988.
- D. H. Ferris and S. Hill, Report NA C45, National
Physical Laboratory, Teddington, 1974.
- R. T. Beaubouef and A. J. Chapman, “Freezing of
Fluids in Forced Flow”, Int. J. Heat Mass
Transfer, vol. 10, pp. 1581-1587, 1967.
- J. L. Duda , M. F. Malone, R. H. Noter, J. S.
Vrentas, “Analysis of Two-Dimensional
Diffusion Controlled Moving Boundary
Problems,” Int. J. Heat Mass Transfer, vol. 18, pp.
901-910, 1975.
- V. Gülkaç, T. Öziş, “On a LOD Method for
Solution of Two-Dimensional Fusion Problem
with Convective Boundary Conditions,”
International Communications in Heat and Mass
Transfer, vol. 31, no.4, pp. 597-606, 2004.
- V. Gülkaç, “On the finite differences schemes for
the numerical solution of two dimensional
moving boundary problem,” Applied
Mathematics and Computation, vol. 168, no.1, pp.
549-556, 2005.
-
- P. Jiang, Z. Ren, “Numerical Investigation of
Forced Convection Heat Transfer in Porous
Media Using a Thermal Non-Equilibrium
Model,” International Journal of Heat and Fluid
Flow, vol. 22, no. 1, pp. 102-110, 2001.
- J. L. Lage, “The Fundamental the Theory of Flow
Through Permeable Media from Darcy to
Turbulence,” Transport Phenomena in Porous
Media (D. M. Ingham & I. Pop. Eds.) Elsevier
Science, Oxford, pp. 1-30, 1998.
- R. Rannacher, “Finite Element Solution of
Diffusion Problems with Irregular Data,” Numer.
Math. vol. 43, no. 2, pp. 308-327, 1976.
- E. M. Sparrow, C. F. Hsu, “Analysis of TwoDimensional
Freezing an Outside of a Coolant
Carying Tube,” Int. J. Heat Mass Transfer, vol.
24, pp. 1345-1357, 1981.
- G. Strang, “On the construction and comparison
of difference schemes,” SIAM J. Numer. Anal.
vol. 5, pp. 506-517, 1968.
- R.J. LeVeque, Finite Volume Methods for
Hyperbolic Problems, in:Cambiridge Texts in
Applied Mathematics. Cambiridge University
Press, Cambridge, UK, 2002.
- R. E. Bank, W.M. Coughan, Jr. W. Fichtner, E. H.
Grosse, D. J. Rose, R.K. Smith, “Transient
Simulation of Silicon Devices and Circuits,”
IEEE Trans. Comput. Aided De sign CAD- vol. 4,
no. 4, pp. 436-450, 1985.