Analysis of the Numerical Solution in the One Dimensional Non-Stationary Diffusion Equation using Schemes of Finite Mimetic Differences and Crank-Nicolson
DOI:
https://doi.org/10.17268/sel.mat.2018.01.10Keywords:
Algorithm in matlab, Mimetic operators, Diffusion equation, Mimetic finite difference method, Crank Nicolson schemeAbstract
The numerical solution of the one-dimensional non-static diffusion equation is proposed, developing an algorithm in software Matlab version 7.0, for which the mimetic finite difference scheme is combined in the approximation of the differential operators of the continuum (gradient and divergence) for the spatial
variable, on a uniform grid, whose discrete differential operators have a second order approximation and the finite difference approach type Crank Nicolson for approximations in the temporary variable.
This proposed algorithm for the mimetic and Crank Nicolson approaches has a better approximation than the finite-difference Crank Nicolson-type scheme.
In addition, the approximation error generated between the numerical solution and the analytical solution is calculated using the maximum standard for the non-stationary diffusion equation with Robin type boundary conditions.
References
Mannarino, I.A. A mimetic finite difference method using Crank Nicolson scheme for unsteady diffusion equation. Revista de matemática: Teoría y Aplicaciones. 2009; 16(2): 221 - 230.
Castillo, J.E. and Grone, R.D. A matrix analysis approach to higher-order aproximations for divergence and gradients satisfying a global conservation law. SIAM Journal on Matrix Analysis and Applications. 2003; 25(1): 128 - 142.
Castillo, J.E. and Yasuda, M. Linear system arising for second order mimetic divergence and gradient operators. Journal of Mathematical Modeling and Algorithm. 2005; 4(1): 67-82.
Guevara- Jordan, J.M.; Rojas, S.; Freites-Villegas, M. and Castillo, J.E. Convergence of a mimetic finite difference method for static diffusion equation. Advances in Difference Equations. 2007; (2007).
Castillo J. and Miranda, G. Mimetic discretization methods. 2013.
Mannarino, I.; Quintana, Y. and Guevara-Jordan, J.M. A numerical study of mimetic scheme for the unsteady heat equation. submitted fo FACYT Review.
Sánchez, E. Finite Differences and Parallel Computing to Simulate Carbon Dioxide Subsurface Mass Transport. 2015.
Sanchez, E.; Paolini, C. and Castillo, J.E. Analyzing Diffusive-Advective-Reactive Processes Using Mimetic Finite Differences with implications in Carbon Dioxide Geologic Storange. 2014.
Gyrya, V.; Koustantin, L. and Manzini, G. The arbitrary order mixed mimetic finite difference method for the diffusion equation. 2016 ; 50(3), 851-877.
Mannarino, I. Un método mimético de diferencias finitas para la ecuación no estática de difusión. Master thesis, Universidad Central de Venezuela, Caracas (2007).
Published
How to Cite
Issue
Section
License
The authors who publish in this journal accept the following conditions:
1. The authors retain the copyright and assign to the journal the right of the first publication, with the work registered with the Creative Commons Attribution License,Atribución 4.0 Internacional (CC BY 4.0) which allows third parties to use what is published whenever they mention the authorship of the work And to the first publication in this magazine.
2. Authors may make other independent and additional contractual arrangements for non-exclusive distribution of the version of the article published in this journal (eg, include it in an institutional repository or publish it in a book) provided they clearly state that The paper was first published in this journal.
3. Authors are encouraged to publish their work on the Internet (for example, on institutional or personal pages) before and during the review and publication process, as it can lead to productive exchanges and to a greater and more rapid dissemination Of the published work.
