I've got quite a long question. I would really appreciate it if you could help me out, because I'm stuck at the moment.
I need to find the order of the local truncation error of Modified Euler, applied to the following system of equations:
\begin{equation}\label{eq:3} \begin{cases} \frac{\partial c}{\partial t} = \frac{\partial^2 c}{\partial x^2} - 1, & 0 < x < 1\\ \frac{\partial c(x=0,t)}{\partial x} = 0, \frac{\partial c(x=1), t}{\partial x} = 0, & t\geq 0\\ c(x,0) = f(x), & 0<x<1 \end{cases} \end{equation}
For the spatial discretization of the initial-boundary-value problem, central, second-order accurate nite- differencing is applied. The mesh size is denoted by $h$ and the resulting semi-discrete system by \begin{equation} \frac{\mbox{d}y}{\mbox{d}t} = Ay + r \end{equation} (From a previous question, we calculated $A$, which has order $h^2$, and $r$.) I have written out M.E. for this equation, with $\tau$ as time step, as follows: \begin{equation*} \begin{split} w^{j+1}&=w^j+\frac{\tau}{2}(Aw^j+r+A(w^j+\tau Aw^j +\tau r)+r)\\ &\mbox{ }...\\ &=w^j + (\tau + \frac{\tau^2}{2}A)(Aw^j + r) \end{split} \end{equation*}
Up to this, I am quite sure that it works out. Yet, from now on the trouble starts. My elaboration on the order of the local truncation error is as follows:
The local truncation error $\theta$ of this method can be computed: \begin{equation}\label{eq:11} \theta^{j+1} = \frac{c^{j+1} - z^{j+1}}{\tau}, \end{equation} where $z^{j+1}$ is computed by applying the Modified Euler method to solution at time $t_j$ (given by $c^j$), hence \begin{equation}\label{eq:12} z^{j+1} = c^j + (\tau + \frac{\tau^2}{2}A)(Ac^j + r^j). \end{equation} Substituting this into the equation for $\theta^{j+1}$ yields: \begin{equation}\label{eq:13} \begin{split} \theta^{j+1} &= \frac{c^{j+1} - c^j - (\tau + \frac{\tau^2}{2}A)(Ac^j +r^j)}{\tau}\\ &= \frac{c^{j+1} - c^j}{\tau} - (1+\frac{\tau}{2}A)(Ac^j + r^j)\\ &= \frac{\partial c^j}{\partial t} + \epsilon^j - (1+\frac{\tau}{2}A)((\frac{\partial^2 c^j}{\partial x^2} - 1) + \mu^j), \end{split} \end{equation} in which \begin{equation*} \epsilon = \frac{\tau}{2}\frac{\partial^2 c}{\partial t^2}(x_i, \zeta_j), \zeta_j \in (t_j, t_{j+1}), \mbox{ and } \mu^j = -\frac{h^2}{12}\frac{\partial^4 c}{\partial x^4}(\xi_i, t_j), \xi \in (x_{i-1}, x_{i+1}) \end{equation*} follow from the truncation errors of the numerical differentiation, equations (3.2) and (3.9) from the book. (These equations were given in my study book).
Hence, we get: \begin{equation*} \begin{split} \theta^{j+1} &= \frac{\partial c^j}{\partial t} + \epsilon^j - (1+\frac{\tau}{2}A)((\frac{\partial^2 c^j}{\partial x^2} - 1) + \mu^j)\\ &= \epsilon^j - \frac{\tau}{2}A\frac{\partial^2c^j}{\partial x^2} - \mu^j - \frac{\tau}{2}A\mu^j, \end{split} \end{equation*} such that the method is of order $O(\tau + h^2)$
Now, I showed this to my professor and he said that the local truncation error should be of order $O(\tau^2 + h^2)$, but he studied this elaboration and could not find a mistake. He calculated the error with a different approach however.
I am wondering if you could help me with this 'solution'; is this a possible approach? And if it is, are there any flaws in it?
If you could help me, I'd be very happy since this problem has been causing me quite a lot of time already and I don't get it.
Thanks in advance!