International Journal of Differential Equations
Volume 2011 (2011), Article ID 520840, 16 pages
http://dx.doi.org/10.1155/2011/520840
Research Article

Solving Advection Equations by Applying the Crank-Nicolson Scheme Combined with the Richardson Extrapolation

Zahari Zlatev,1 Ivan Dimov,2 István Faragó,3 Krassimir Georgiev,2 Ágnes Havasi,4 and Tzvetan Ostromsky2

1National Environmental Research Institute, Aarhus University, DK-4000 Roskilde, Denmark
2Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, BG-1113 Sofia, Bulgaria
3Department of Applied Analysis and Computational Mathematics, Eötvös Loránd University, H-1117 Budapest, Hungary
4Department of Meteorology, Eötvös Loránd University, H-1117 Budapest, Hungary

Received 23 May 2011; Revised 3 August 2011; Accepted 15 September 2011

Academic Editor: Roderick Melnik

Copyright © 2011 Zahari Zlatev et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Advection equations appear often in large-scale mathematical models arising in many fields of science and engineering. The Crank-Nicolson scheme can successfully be used in the numerical treatment of such equations. The accuracy of the numerical solution can sometimes be increased substantially by applying the Richardson Extrapolation. Two theorems related to the accuracy of the calculations will be formulated and proved in this paper. The usefulness of the combination consisting of the Crank-Nicolson scheme and the Richardson Extrapolation will be illustrated by numerical examples.