I am looking for help on this proof of a maximum principle for the discrete heat equation. The following is from Introduction to Partial Differential Equations (Tveito, Winther).
Consider the Heat Equation
with an explicit finite difference scheme that can be expressed as
where $t_m = m \Delta t, \ x_j = j\Delta x$, and $v(x_j,t_m)$ is an approximation to $u(x_j,t_m)$.
Define the rectangle
and the associated "lower boundary"
Let
Want to prove
Condition (6.22) is $1-2r \geq 0$
I am having some trouble understanding the proof of this theorem. It goes like this (upper bound only):
Assume $1-2r \geq 0$. Fix a time level $t_m$. Assume $v_j^m \leq V^+$ for $j = 1,\ldots,n$.
We then have
$v_j^{m+1} = rv_{j-1}^m + (1-2r)v_j^m + rv_{j+1}^m, \ j = 1,\ldots, n$.
Thus
$v_j^{m+1} \leq rV^+ + (1-2r)V^+ + rV^+ = V^+, \ j = 1,\ldots,n$.
Since this holds for any $j = 0,\ldots, n+1$, the result follows by induction on $m$.
Thought: We know that it holds for $m = 0, \ j = 1,\ldots,n$, due to the initial condition. We then assume it holds for an arbitrary $m$ and $j = 1,\ldots,n$. We then prove that it holds for $m+1$, and thus we are done, since the boundary conditions imply that it holds for $j = 0$ and $j = n+1$ as well.
Is this a correct interpretation? I am familiar with induction, but I got a little confused when we're treating both $m$ and $j$ at the same time, sort of speak.