Computational Analysis of the Stability of 2D Heat Equation on Elliptical Domain Using Finite Difference Method
Asian Research Journal of Mathematics,
Page 819
DOI:
10.9734/arjom/2020/v16i330177
Abstract
Aims: The aim and objective of the study to derive and analyze the stability of the finite difference schemes in relation to the irregularity of domain.
Study Design: First of all, an elliptical domain has been constructed with the governing two dimensional (2D) heat equation that is discretized using the Finite Difference Method (FDM). Then the stability condition has been defined and the numerical solution by writing MATLAB codes has been obtained with the stable values of time domain.
Place and Duration of Study: The work has been jointly conducted at the MUET, Jamshoro and QUEST, Nawabshah Pakistan from January 2019 to December 2019.
Methodology: The stability condition over an elliptical domain with the nonuniform step size depending upon the boundary tracing function is derived by using Von Neumann method.
Results: From the results it was revealed that stability region for the small number of mesh points remains larger and gets smaller as the number of mesh nodes is increased. Moreover, the ranges for the time steps are defined for varied spatial step sizes that help to find the stable solution.
Conclusion: The corresponding stability range for number of nodes N=10, 20, 30, 40, 50, and 60 was found respectively. Within this range the solution remains smooth as time increases. The results of this study attempt to provide the stable solution of partial differential equations on irregular domains.
Keywords:
 Modeling and simulation
 stability analysis
 computational analysis
 finite difference method
 elliptical domain
 heat equation.
How to Cite
References
[Retrieved in May 2016]
OlsenKettle L. Numerical solution of partial differential equations; 2011.
Isaacson E, Keller HB. Analysis of Numerical Methods. 1994;541.
Han Chen, Cho Hong Min, Frederick Gibou. A Supraconvergent finite difference scheme for the poisson and heat equations on irregular domains and NonGraded Adaptive Grids; 2006.
Peter McCorquodale, Phillip Colella, Hans Johansen, A cartesian grid embedded boundary method for the heat equation on irregular domains.
DOI: 10.1006/jcph.2001.6900
Available:http://www.idealibrary.com
Verena Hora, Peter Gruber. Parallel Numerical Solution of 2D Heat Equation, Parallel Numerics. 2005;4756.
Gerald W. Recktenwald. Finitedifference approximations to the heat equation; 2011.
Kazufumi Ito, Zhilin Li, Yaw Kyei. Higherorder, Cartesian grid based finite difference schemes for elliptic equations on irregular domains. SIAM Journal on Scientific Computing. 2005;27(1): 346367.
Izadian J, Karamooz N. New method for solving poisson equation on irregular domains. Applied Mathematical Sciences. 2012;6(8):369–380.
Gavete L, Benito JJ, Urena F. Generalized finite differences for solving 3D elliptic and parabolic equations. Applied Mathematical Modelling. 2016;40(2):955965.
Gilberto E. Urroz. Convergence, stability, and consistency of finite difference schemes in the solution of partial differential equations; 2004.
Courant R, Friedrichs K, Lewy H. Über die partiellen Differenzengleichungen der mathematischen Physik", Mathematische Annalen (in German). 1928;100(1):3274.
Courant R, Friedrichs K, Lewy H. On the partial difference equations of mathematical physics. AEC Research and Development Report, NYO7689, New York: AEC Computing and Applied Mathematics; 1956.
Courant R, Friedrichs K, Lewy H. On the partial difference equations of mathematical physics. IBM Journal of Research and Development. 1928;11(2):21534.
Charney JG, Fjörtoft R, Von Neumann J. Numerical integration of the barotropic vorticity equation. In The Atmosphere—A Challenge. American Meteorological Society, Boston, MA. 1990;267284.
Lax PD, Richtmyer RD. Survey of the stability of linear finite difference equations. Comm. Pure Appl. Math. 1956;9:267293 MR 79204.
DOI: 10.1002/cpa.3160090206
Crank J, Nicolson P. A practical method for numerical evaluation of solutions of partial differential equations of heat conduction type. Proc. Camb. Phil. Soc. 1947;43.
Yuste SB. Weighted average finite difference methods for fractional diffusion equations. Journal of Computational Physics. 2006;216:264–274.
Yuste SB, Acedo L. An explicit finite difference method and a new von neumanntype stability analysis for fractional diffusion equations. SIAM Journal on Numerical Analysis. 2005;42(5):18621874.
Bilbao S. Modeling of complex geometries and boundary conditions in finite difference/finite volume time domain room acoustics simulation. In IEEE Transactions on Audio, Speech, and Language Processing. 2013;21(7):1524153.
Jincheng Ren, Zhizhong Sun, Xuan Zhao. Compact difference scheme for the fractional subdiffusion equation with Neumann boundary conditions. Journal of Computational Physics. 2013; 232(1):456467.
[ISSN 00219991]
Available:http://dx.doi.org/10.1016/j.jcp.2012.08.026
Ercília Sousa, Can Li. A weighted finite difference method for the fractional diffusion equation based on the Riemann–Liouville derivative. Applied Numerical Mathematics. 2015;90:2237.
[ISSN 01689274]
Avilable:http://dx.doi.org/10.1016/j.apnum.2014.11.007
Ercília Sousa. A second order explicit finite difference method for the fractional advection diffusion equation, Computers & Mathematics with Applications. 2012;64(10):31413152.
[ISSN 08981221]
Available:http://dx.doi.org/10.1016/j.camwa.2012.03.002
Sweilam NH, Khader MM, Mahdy AMS. Cranknicolson finite difference method for solving timefractional diffusion equation. Journal of Fractional Calculus and Applications. 2012;2(2):19.
QuintanaMurillo J, Yuste SB. A finite difference method with nonuniform time steps for fractional diffusion and diffusionwave equations. The European Physical Journal Special Topics. 2013;222(8): 1987–1998.
Norihiro Watanab, Olaf Kolditz. Numerical stability analysis of twodimensional solute transport along a discrete fracture in a porous rock matrix, Water Resources Research. 2015;51(7):5855–5868.
Jiequan Li, Zhicheng Yang. The von Neumann analysis and modified equation approach for finite difference schemes. Applied Mathematics and Computation. 2013;225:610621.
[ISSN 00963003]
Available:http://dx.doi.org/10.1016/j.amc.2013.09.046
Ehlers W, Zinatbakhsh S, Markert B. Stability analysis of finite difference schemes revisited: A study of decoupled solution strategies for coupled multifield problems. Int. J. Numer. Meth. Engng. 2013; 94:758–78.
Baudouin L, Seuret A, Gouaisbaut F, Dattas M. Lyapunov stability analysis of a linear system coupled to a heat equation. IFACPapersOnLine. 2017;50(1):1197811983.
Konangi S, Palakurthi NK, Ghia U. Von Neumann stability analysis of firstorder accurate discretization schemes for onedimensional (1D) and twodimensional (2D) fluid flow equations. Computers & Mathematics with Applications. 2018;75(2):643665.
Jahangir K, Rehman SU, Ahmad F, Pervaiz A. SixthOrder Stable Implicit Finite Difference Scheme for 2D Heat Conduction Equation on Uniform Cartesian Grids with Dirichlet Boundaries. Punjab Univ. J. Math. 2019;51(5):2742.
Singh AK, Bhadauria BS. Finite difference formulae for unequal subintervals using Lagrange’s interpolation formula. Int. J. Math. Anal. 2009;3(17):815.

Abstract View: 2560 times
PDF Download: 2026 times