Research ArticleOpen AccessGoogle Scholar indexed
A Comparative Study of Two Spatial Discretization Schemes for Advection Equation
Department of Mathematics, Faculty of Science, Al Faisaliah Campus, King Abdulaziz University, Jeddah, Saudi Arabia
- 1 Department of Mathematics, Faculty of Science, Al Faisaliah Campus, King Abdulaziz University, Jeddah, Saudi Arabia
International Journal of Modern Nonlinear Theory and Application·Volume 05 (2016)·Pages 59–66·Published 2 March 2016·DOI10.4236/ijmnta.2016.51006
Copy link · social · email
Abstract
In this paper, we describe a comparison of two spatial discretization schemes for the advection equation, namely the first finite difference method and the method of lines. The stability of the methods has been studied by Von Neumann method and with the matrix analysis. The methods are applied to a number of test problems to compare the accuracy and computational efficiency. We show that both discretization techniques approximate correctly solution of advection equation and compare their accuracy and performance.
KeywordsAdvection EquationFinite Difference MethodThe Method of LinesVon Neumann Method
- Biazar, J., Ghazvini, H. and Eslami, M. (2009) He’s Homotopy Perturbation Method for Systems of Integro-Differential Equations. Chaos, Solitons & Fractals, 39, 1253-1258. http://dx.doi.org/10.1016/j.chaos.2007.06.001
- Biazar, J. and Eslami, M. (2011) Modified HPM for Solving Systems of Volterra Integral Equations of the Second Kind. Journal of King Saud University—Science, 23, 35-39. http://dx.doi.org/10.1016/j.jksus.2010.06.004
- Biazar, J., Eslami, M. and Islam, M.R. (2012) Differential Transform Method for Special Systems of Integral Equations. Journal of King Saud University—Science, 24, 211-214. http://dx.doi.org/10.1016/j.jksus.2010.08.015
- Eslami, M. (2014) New Homotopy Perturbation Method for a Special Kind of Volterra Integral Equations in Two-Dimensional Space. Computational Mathematics and Modeling, 25, 135-148. http://dx.doi.org/10.1007/s10598-013-9214-x
- Biazar, J. and Eslami, M. (2011) A New Homotopy Perturbation Method for Solving Systems of Partial Differential Equations. Computers & Mathematics with Applications, 62, 225-234. http://dx.doi.org/10.1016/j.camwa.2011.04.070
- Biazar, J., Eslami, M. and Aminikhah, H. (2009) Application of Homotopy Perturbation Method for Systems of Volterra Integral Equations of the First Kind. Chaos, Solitons & Fractals, 42, 3020-3026. http://dx.doi.org/10.1016/j.chaos.2009.04.016
- Biazar, J. and Eslami, M. (2012) A New Method for Solving the Hyperbolic Telegraph Equation. Computational Mathematics and Modeling, 23, 519-527. http://dx.doi.org/10.1007/s10598-012-9153-y
- Wazwaz, A.M. (2009) Partial Differential Equations and Solitary Waves Theory. Higher Education Press, Beijing and Springer-Verlag, Berlin Heidelberg. http://dx.doi.org/10.1007/978-3-642-00251-9
- George, K. and Twizell, E.H. (2006) Stable Second-Order Finite-Difference Methods for Linear Initial-Boundary-Value Problems. Applied Mathematics Letters, 19, 146-154. http://dx.doi.org/10.1016/j.aml.2005.04.003
- Smith, G.D. (1985) Numerical Solution of Partial Differential Equations (Finite Difference Method). 3rd Edition, Oxford University Press, Oxford.
- Al-Malki, N.A. and Bakodah, H.O. (2014) A Stable Difference Algorithm for the Solution of Advection Equation. Far East Journal of Applied Mathematics, 88, 139-149.
- Sharaf, A.A. and Bakodah, H.O. (2005) A Good Spatial Discretization in the Method of Lines. Applied Mathematics and Computation, 171-172, 1253-1263. http://dx.doi.org/10.1016/j.amc.2005.01.144