Numerical Modelling of Nonlinear Diffusion Phenomena on a Sphere - Simulation and Modeling Methodologies, Technologies and Application - page 65

spatial approximation of the numerical solution. The maximum relative errors on the
chosen grids are 2 . 20 · 10 − 2 , 8 . 91 · 10 − 3 and 1 . 38 · 10 − 3 , respectively.
In Fig. 10 we also plot the graph of δ n in time on the grid 6 ◦ × 6 ◦ . According to the
periodical character of analytical solution (32) determined by forcing (34), the relative
error periodically grows and decays too.
Acknowledgements. This research was partially supported by grants of the National
System of Researchers (SNI) of Mexico, No. 14539 and 26073, and is part of the project
PAPIIT-UNAM IN104811, Mexico.
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Appendix
As it was noticed in Section 2, this is exactly the dimensional splitting that allows
employing periodic boundary conditions in both spatial coordinates, thereby providing
a fast and simple numerical algorithm for finding the solution [5].
A not less important benefit of the coordinate splitting is that one may involve spa-
tial stencils of high-order approximations without complicating the computer code. In
particular, in [6], for the linear diffusion equation, we tested fourth-order schemes. The
limits on the size of the present paper do not permit us to make a comprehensive anal-
ysis of the fourth-order schemes in the nonlinear case. This is a subject of a separate
study. However, the approach is similar to [6], since due to (7) the nonlinear term gets
linearised and the problem reduces to the linear case.