Linear manufactured solution

$u^n = ct_n+I$ fulfills the discrete equations!

First, $$\begin{align} \lbrack D_t^+ t\rbrack^n &= \frac{t_{n+1}-t_n}{\Delta t}=1, \tag{34}\\ \lbrack D_t^- t\rbrack^n &= \frac{t_{n}-t_{n-1}}{\Delta t}=1, \tag{35}\\ \lbrack D_t t\rbrack^n &= \frac{t_{n+\half}-t_{n-\half}}{\Delta t}=\frac{(n+\half)\Delta t - (n-\half)\Delta t}{\Delta t}=1\tag{36} \end{align}$$

Forward Euler: $$[D^+ u = -au + b]^n$$

$a^n=a(t_n)$, $b^n=c + a(t_n)(ct_n + I)$, and $u^n=ct_n + I$ results in $$c = -a(t_n)(ct_n+I) + c + a(t_n)(ct_n + I) = c$$