equation in (\[eq:yoshidaB\]) is awkward to solve, we use an Euler estimate in one of the terms. Here is one integration time step of (\[eq:yoshida\]) in discrete semi-implicit form:

\[eq:discreteyoshida\] $$\begin{aligned}
\left( 1 - \frac{k}{4} A_{x} \right) u^{\star} & = \left( 1 +
\frac{k}{4} A_{x} \right) u(t) \label{eq:discreteyoshidaA} \\ u_{E} &=
u^{\star} + k A_{y} u^{\star} + h (1-u^{\star})
u^{\star} \label{eq:discreteyoshidaB} \\ \left( 1 - \frac{k}{2} A_{y}
- \frac{h}{2} (1-u_{E}) \right) u^{\star \star} & = \left( 1 +
\frac{k}{2} A_{y} + \frac{h}{2} (1-u^{\star}) \right) u^{\star}
\label{eq:discreteyoshidaC} \\
\left( 1 - \frac{k}{4} A_{x} \right) u(t+h) & = \left( 1 + \frac{k}{4}
A_{x} \right) u^{\star \star}.
\label{eq:discreteyoshidaD}\end{aligned}$$

Equations (\[eq:discreteyoshidaA\]), (\[eq:discreteyoshidaC\]), and (\[eq:discreteyoshidaD\]) are solved in sequence as multiple independent tridiagonal systems for $u^{\star}, u^{\star \star}$ and the final step $u(t+h)$.

Symmetries in 2-D case
----------------------

Our Godunov scheme (\[eq:discreteyoshida\]) is not rotationally symmetric, and thus one way to estimate the error is to assess a solution using (\[eq:discreteyoshida\]) for a symmetric problem. Again, we can use `pdepe` but now with the cylindrically symmetric parameter choice `m`=1 (see Section 12.5 in [@higham2005]). Figure \[fig:2dpcolorplots\] shows that any asymmetries are not apparent without more careful examination. Even the wave front portrait of the 2-D 