Numerical Approach for the two-dimensional heat equation problem with convective boundary conditions

Year 2017, Volume: 21 Issue: 3, 343 - 349, 01.06.2017

Vildan Gülkaç

https://doi.org/10.16984/saufenbilder.283775

Abstract

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.  

Keywords

Flux limiters, LOD method, ADI method, moving boundary problems, Strang splitting

İki-Boyutlu Konvektif Sınır Koşullu Erime Problemi İçin Nümerik Yaklaşım

Abstract

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.  

Keywords

Flux Limiters, LOD metodu, hareketli sınır problemleri, Strang splitting

References

• 1. T. Öziş , V. Gülkaç , Application of variable interchange method for solution of two-dimensional fusion problem with convective boundary conditions., Numerical Heat Transfer, Part A, 44, 85-95, 2003.
• 2. V. Gülkaç, A Numerical Solution of Two-Dimensional Fusion Problem with Convective Boundary Conditions, International Journal for Computational Methods in Engineering Science and Mechanics, 11, 20-26, 2010.
• 3. J. Crank, Numerical Methods in Heat Transfer, John Wiley, 1981.
• 4. 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.
• 5. H. Rasmussen, An Approximate Method for Solving Two-Dimensional Stefan Problems., Lett. Heat Mass Transfer, 4, 273-277, 1997.
• 6. C. W. Cryer, A Survey of Steady-State Porous Flow Free Boundary Problems, MRC Tech. Summary Report 1657, University of Wisconsin, Madison, 1976.
• 7. 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.
• 8. R. M. Furzeland , Symposium on Free and Moving Boundary Problems in Heat Flow and Diffusion, Bull. Inst. Maths Applics, 15, 172-176, 1979.
• 9. J. M. Aitchison, Numerical Treatment of a Singularity in a Free Boundary Problem, Proceedings of the Royal Society of London, Series A, 573-580, 1972.
• 10. J. M. Aitchison, The Numerical Solution of a Minimization Problem Associated with a Free Surface Flow., J. Inst. Maths Applics, 20, 33-44, 1977.
• 11. H. G. Landau, Heat Conduction in a Melting Solid, Qart. Appl. Math., 8, 81-94, 1950.
• 12. R. S. Gupta, A. Kumar, Isotherm Migration Method Applied to Fusion Problems with Convective Boundary Conditions. Int. J. Numer. Meth. Eng. 26, 2547-2558, 1988.
• 13. D. H. Ferris and S. Hill, Report NA C45, National Physical Laboratory, Teddington, 1974.
• 14. R. T. Beaubouef and A. J. Chapman, Freezing of Fluids in Forced Flow, Int. J. Heat Mass Transfer, 10, 1581-1587, 1967.
• 15. 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, 18, 901-910, 1975
• 16. V. Gülkaç, T. Öziş, Erratum to “On a LOD Method for Solution of Two-Dimensional Fusion Problem with Convective Boundary Conditions”, International Communications in Heat and Mass Transfer, 31, 1233, 2004.
• 17. V. Gülkaç, On the finite differences schemes for the numerical solution of two dimensional moving boundary problem, Applied Mathematics and Computation, 168, 549-556, 2005.
• 18. W. J. Minkowycz and E. M. Sparrow, Local Non-Similar Solutions for Natural Convection on a Vertical Cylinder, Journal of Heat Transfer, 96 (2), 178-183, 1974.
• 19. 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, 22(1), 102-110, 2001.
• 20. 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, 1-30, 1998.
• 21. R. Rannacher, Finite Element Solution of Diffusion Problems with Irregular Data. Numer. Math. 43 (2), 308-327, 1976.
• 22. E. M. Sparrow, C. F. Hsu, Analysis of Two-Dimensional Freezing an Outside of a Coolant Carying tube. Int. J. Heat Mass Transfer, 24, 1345-1357, 1981.
• 23. G. Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal. 5, 506-517, 1968.
• 24. R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, in:Cambiridge Texts in Applied Mathematics. Cambiridge University Press, Cambiridge, UK, 2002.
• 25. 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-4(4) 436-450, 1985.