# Discrete maximum principle for finite difference to the heat equation

Given a continuous-time, discrete space approximation of $u_t = u_{xx}$ with homogeneous Dirichlet boundary condition. The discrete ODE version for the nodal values is: $u'_j = \frac{u_{j-1}+u_{j+1} - 2u_j}{(\delta x)^2}$ for $j = 1,\dots, N-1$ over the domain $(0, N\delta x)$ where $\delta x = 1/N$. Note that $u_j (t) = u(j\delta x, t)$, and $u_0(t) = u_N(t) = 0$.

Question: Prove the discrete maximum principle, which is: If $w_j(t)$ for $j = 1,\dots, N-1$ solves the ODE above and $w_0(t) = w_N(t) = 0$, then max$_{j, t} w_j(t)$ and min$_{j, t} w_j(t)$ are either achieved at $t= 0$ or $j = 0$ or $j=N$.

My progress: I have tried using contradiction method to mimic the proof in the continuous case for the PDE: $u_t - \Delta u = 0$, but failed to reach anywhere. Can anyone please help me with this problem?

• @Byron Schmuland: thank you very much for your help on editing. Can you give this question a try? – user177196 Oct 14 '15 at 21:36
• I'm thinking about it, but I'm a bit puzzled by the notation and terminology. In particular, why is this an ODE when the continuous version is a PDE? – user940 Oct 14 '15 at 21:41
• The problem used central difference for the term $u_{xx}$, while it does nothing for the term $u'_j$. So that's why you have a PDE. Keep in mind that the discretization is only in space, but not in time (so no approximation is done with respect to $u_t$). – user177196 Oct 14 '15 at 21:46
• Can anyone give me some help on this problem? – user177196 Oct 15 '15 at 18:39