namely only information from the previous step, $u(t)$. Euler estimate $u_{E}$ is an explicit one step computation using $u(t)$. Significant advantages are: the matrix on the left hand side is tridiagonal with constants on the sub/super-diagonals, and the diagonal terms are $O(1)$ strong. The procedure (\[eq:semidifference\]) is only linearly stable but we will show empirically that it gives good results when compared to `pdepe` when this `MatLab` function is appropriate, that is, in both the one-dimensional and rotationally symmetric 2-D case. Not only is the method (\[eq:semidifference\]) step-wise stable but also stable for initial data which may not be smooth.

Figure \[fig:1dplots\] shows the results for $h=1/5$, $k=2.5$ compared to `pdepe`. Notice that at $t=20$ the agreement is remarkable; and that at $t=h$, where the wave front profile is very steep, our Godunov splitting described in Section \[sec:Godunov2Dsplitting\], specifically eq.(\[eq:yoshida\]), is very stable. The CFL number, $k=2.5$, used to get Figure \[fig:1dplots\] is much larger than would be possible with an explicit method [@leveque2007].

2-D case: Godunov–Strang–Yoshida splittings {#sec:Godunov2Dsplitting}
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It turns out that a generalization to the 2-D problem is a straightforward variant of Strang–Yoshida splittings [@strang1968; @yoshida1990], which are themselves variants of Godunov’s method [@godunov1987]. The following is 