semi-implicit Godunov scheme for the Fisher/KPP equation with a constant carrying capacity $\mathcal{K}$, described in Section \[sec:methodsandsplittings\]. In one dimension, the expected traveling wave [@fisher1937; @kolmogorov1937] develops as shown in Figure \[fig:1dplots\]. In other tests, not shown here, we saw that almost any concentrated inital condition will develop similar waves: for example, two nearby peaks. Our scheme is on a rectangular grid, so in 2-D we need to ensure that in cylindrically symmetric situations we can control the errors due to the $x - y$ asymmetry. The error plots, Figure \[fig:symmetryornot\], show that we can reduce the asymmetries somewhat by an alternating direction scheme. In any case, these errors are very small even for a relatively high CFL number. As in the one-dimensional situation, a traveling wave also develops as expected [@kolmogorov1937]: Figure \[fig:2dpcolorplots\] and Figure \[fig:2Dwavefrontanderrors\] show this development and compare the results to the 2-D symmetric version of the Matlab function `pdepe`. Because the $\mathbf{x}$ discretization has truncation errors proportional to $(\Delta x)^{3}$, after $O(\frac{1}{h})$ time steps we should not be surprised to see the errors shown in Figure \[fig:2Dwavefrontanderrors\] turn up for very small $h$ and behave roughly as $O(\frac{1}{h} (\Delta x)^{3})$.

In Secs. \[sec:fisherkppvarK\] we extended 