case in the left-hand panel of Figure \[fig:2Dwavefrontanderrors\] and the error estimate in the right-hand panel of the same Figure are not sufficiently quantitative. In particular, there should be no distinction between $x$ and $y$ directions in (\[eq:yoshida\]), while a 2-D plot of the error distribution shows a small asymmetry (compare the right-hand plot in Figure \[fig:symmetryornot\] to the left).

For this reason, we implemented an alternating direction method, [ *à la*]{} Crank-Nicholson [@leveque2007; @ritchmyer1967], which makes the error distribution more symmetric. The left panel of Figure \[fig:symmetryornot\] shows that, while symmetrization only slightly improves the r.m.s. and maximum errors, they have now the desirable property of being more rotationally equi-distributed: respectively, cyclic groups $C_{4}$ vs. $C_{2}$. Relatively larger deviations from the `pdepe` solution now correspond to directions diagonal to the spatial lattice, as expected, and do not reflect the arbitrary choice of $x$ and $y$ in the integration.

Fisher/KPP on maps {#sec:fisherkpp}
==================

The next natural step when applying a reaction-diffusion equation to the modeling of population dispersal is to include geographical and environmental effects. In this Section, we discuss how to implement our solver on domains with space- and time-dependent $\mathcal K$, and then how to treat irregular 