Advances in Mathematical Physics
Volume 2009 (2009), Article ID 568296, 23 pages
Research Article

High-Order Compact Implicit Difference Methods For Parabolic Equations in Geodynamo Simulation

1Program of Mathematics and Statistics, Louisiana Tech University, Ruston, LA 71270, USA
2Planetary Geodynamics Laboratory, Code 698, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
3Joint Center for Earth Systems Technology, University of Maryland Baltimore County, Baltimore, MD 21228, USA

Received 8 January 2009; Accepted 26 March 2009

Academic Editor: Jan Hesthaven

Copyright © 2009 Don Liu 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.


A series of compact implicit schemes of fourth and sixth orders are developed for solving differential equations involved in geodynamics simulations. Three illustrative examples are described to demonstrate that high-order convergence rates are achieved while good efficiency in terms of fewer grid points is maintained. This study shows that high-order compact implicit difference methods provide high flexibility and good convergence in solving some special differential equations on nonuniform grids.